Physlib.SpaceAndTime.Space.ConstantSliceDist — Physlib

Physlib
SpaceAndTime
Space
ConstantSliceDist

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 Mathlib.Analysis.Calculus.ContDiff.FiniteDimension
public import Physlib.SpaceAndTime.Space.Derivatives.Basic
public import Physlib.SpaceAndTime.Space.Slice
public import Mathlib.Analysis.Calculus.ParametricIntegral

Constant slice distributions

i. Overview

In this module we define the lift of distributions on Space d to distributions on Space d.succ which are constant between slices in the ith direction.

This is used, for example, to define distributions which are translationally invariant in the ith direction.

Examples of distributions which can be constructed in this way include the dirac deltas for lines and planes, rather then points.

ii. Key results

sliceSchwartz : The continuous linear map which takes a Schwartz map on Space d.succ and gives a Schwartz map on Space d by integrating over the ith direction.

constantSliceDist : The distribution on Space d.succ formed by a distribution on Space d which is translationally invariant in the ith direction.

iii. Table of contents

A. Schwartz maps

A.1. Bounded condition for derivatives of Schwartz maps on slices

A.2. Integrability for of Schwartz maps on slices

A.3. Continiuity of integrations of slices of Schwartz maps

A.4. Derivative of integrations of slices of Schwartz maps

A.5. Differentiability as a slices of Schwartz maps

A.6. Smoothness as slices of Schwartz maps

A.7. Iterated derivatives of integrations of slices of Schwartz maps

A.8. The map integrating over one component of a Schwartz map

B. Constant slice distribution

B.1. Derivative of constant slice distributions

iv. References

@[expose] public section

A. Schwartz maps

A.1. Bounded condition for derivatives of Schwartz maps on slices



lemma schwartzMap_slice_bound {n m} {d : ℕ} (i : Fin d.succ) :
    ∃ rt, ∀ (η : 𝓢(Space d.succ, ℝ)), ∃ k,
    Integrable (fun x : ℝ => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    (∀ (x : Space d), ∀ r, ‖(slice i).symm (r, x)‖ ^ m *
      ‖iteratedFDeriv ℝ n ⇑η ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ (rt)‖⁻¹)
      ∧ k = (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup
        fun m => SchwartzMap.seminorm ℝ m.1 m.2) η) := byn:ℕm:ℕd:ℕi:Fin d.succ⊢ ∃ rt,
  ∀ (η : 𝓢(Space d.succ, ℝ)),
    ∃ k,
      Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
        (∀ (x : Space d) (r : ℝ),
            ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
          k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η
  obtain ⟨rt, hrt⟩ : ∃ r, Integrable (fun x : ℝ => ‖((1 + ‖x‖) ^ r)⁻¹‖) volume := byn:ℕm:ℕd:ℕi:Fin d.succ⊢ ∃ r, Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖) volumen:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume⊢ ∃ rt,
  ∀ (η : 𝓢(Space d.succ, ℝ)),
    ∃ k,
      Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
        (∀ (x : Space d) (r : ℝ),
            ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
          k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η
      obtain ⟨r, h⟩ := Measure.HasTemperateGrowth.exists_integrable (μ := volume (α := ℝ))n:ℕm:ℕd:ℕi:Fin d.succr:ℕh:Integrable (fun x => (1 + ‖x‖) ^ (-↑r)) volume⊢ ∃ r, Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖) volumen:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume⊢ ∃ rt,
  ∀ (η : 𝓢(Space d.succ, ℝ)),
    ∃ k,
      Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
        (∀ (x : Space d) (r : ℝ),
            ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
          k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η
      use rhn:ℕm:ℕd:ℕi:Fin d.succr:ℕh:Integrable (fun x => (1 + ‖x‖) ^ (-↑r)) volume⊢ Integrable (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖) volumen:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume⊢ ∃ rt,
  ∀ (η : 𝓢(Space d.succ, ℝ)),
    ∃ k,
      Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
        (∀ (x : Space d) (r : ℝ),
            ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
          k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η
      convert h using 1n:ℕm:ℕd:ℕi:Fin d.succr:ℕh:Integrable (fun x => (1 + ‖x‖) ^ (-↑r)) volume⊢ (fun x => ‖((1 + ‖x‖) ^ r)⁻¹‖) = fun x => (1 + ‖x‖) ^ (-↑r)n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume⊢ ∃ rt,
  ∀ (η : 𝓢(Space d.succ, ℝ)),
    ∃ k,
      Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
        (∀ (x : Space d) (r : ℝ),
            ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
          k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η
      funext xn:ℕm:ℕd:ℕi:Fin d.succr:ℕh:Integrable (fun x => (1 + ‖x‖) ^ (-↑r)) volumex:ℝ⊢ ‖((1 + ‖x‖) ^ r)⁻¹‖ = (1 + ‖x‖) ^ (-↑r)n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume⊢ ∃ rt,
  ∀ (η : 𝓢(Space d.succ, ℝ)),
    ∃ k,
      Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
        (∀ (x : Space d) (r : ℝ),
            ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
          k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η
      simp only [norm_inv, norm_pow, Real.norm_eq_abs, Real.rpow_neg_natCast, zpow_neg,
        zpow_natCast, inv_inj]n:ℕm:ℕd:ℕi:Fin d.succr:ℕh:Integrable (fun x => (1 + ‖x‖) ^ (-↑r)) volumex:ℝ⊢ |1 + |x|| ^ r = (1 + |x|) ^ rn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume⊢ ∃ rt,
  ∀ (η : 𝓢(Space d.succ, ℝ)),
    ∃ k,
      Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
        (∀ (x : Space d) (r : ℝ),
            ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
          k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η
      rw [abs_of_nonneg (byn:ℕm:ℕd:ℕi:Fin d.succr:ℕh:Integrable (fun x => (1 + ‖x‖) ^ (-↑r)) volumex:ℝ⊢ 0 ≤ 1 + |x|n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume⊢ ∃ rt,
  ∀ (η : 𝓢(Space d.succ, ℝ)),
    ∃ k,
      Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
        (∀ (x : Space d) (r : ℝ),
            ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
          k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η positivityAll goals completed! 🐙n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume⊢ ∃ rt,
  ∀ (η : 𝓢(Space d.succ, ℝ)),
    ∃ k,
      Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
        (∀ (x : Space d) (r : ℝ),
            ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
          k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)]n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume⊢ ∃ rt,
  ∀ (η : 𝓢(Space d.succ, ℝ)),
    ∃ k,
      Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
        (∀ (x : Space d) (r : ℝ),
            ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
          k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume⊢ ∃ rt,
  ∀ (η : 𝓢(Space d.succ, ℝ)),
    ∃ k,
      Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
        (∀ (x : Space d) (r : ℝ),
            ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
          k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η
  use rthn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume⊢ ∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η
  intro ηhn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)⊢ ∃ k,
  Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    (∀ (x : Space d) (r : ℝ),
        ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
      k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η
  have h0 := one_add_le_sup_seminorm_apply (𝕜 := ℝ) (m := (rt + m, n))
      (k := rt + m) (n := n) le_rfl (le_rfl) ηhn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space d.succ),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∃ k,
  Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    (∀ (x : Space d) (r : ℝ),
        ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
      k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η
  simp at h0hn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∃ k,
  Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    (∀ (x : Space d) (r : ℝ),
        ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
      k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η
  let k := 2 ^ (rt + m, n).1 *
    ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηhn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∃ k,
  Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    (∀ (x : Space d) (r : ℝ),
        ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
      k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η
  refine ⟨k, ⟨hrt, fun x r => ?_, byn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η rflAll goals completed! 🐙⟩⟩
  trans k * ‖(1 + ‖((slice i).symm (r, x))‖) ^ rt‖⁻¹n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤
  k * ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖⁻¹n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ k * ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖⁻¹ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹; swapn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ k * ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖⁻¹ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤
  k * ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖⁻¹
  ·n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ k * ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖⁻¹ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹ refine mul_le_mul_of_nonneg (byn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ k ≤ k rflAll goals completed! 🐙) ?_ (byn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ 0 ≤ k positivityAll goals completed! 🐙) (byn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ 0 ≤ ‖(1 + ‖r‖) ^ rt‖⁻¹ positivityAll goals completed! 🐙)
    by_cases rt = 0posn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:rt = 0⊢ ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖⁻¹ ≤ ‖(1 + ‖r‖) ^ rt‖⁻¹negn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:¬rt = 0⊢ ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖⁻¹ ≤ ‖(1 + ‖r‖) ^ rt‖⁻¹
    ·posn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:rt = 0⊢ ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖⁻¹ ≤ ‖(1 + ‖r‖) ^ rt‖⁻¹ subst rtposn:ℕm:ℕd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)x:Space dr:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ 0)⁻¹‖) volumeh0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (0 + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (0 + m) * ((Finset.Iic (0 + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (0 + m, n).1 * ((Finset.Iic (0 + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ‖(1 + ‖(slice i).symm (r, x)‖) ^ 0‖⁻¹ ≤ ‖(1 + ‖r‖) ^ 0‖⁻¹
      simpAll goals completed! 🐙
    refine inv_anti₀ ?_ ?_neg.refine_1n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:¬rt = 0⊢ 0 < ‖(1 + ‖r‖) ^ rt‖neg.refine_2n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:¬rt = 0⊢ ‖(1 + ‖r‖) ^ rt‖ ≤ ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖
    ·neg.refine_1n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:¬rt = 0⊢ 0 < ‖(1 + ‖r‖) ^ rt‖ simp only [norm_eq_abs, norm_pow]neg.refine_1n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:¬rt = 0⊢ 0 < |1 + |r|| ^ rt
      rw [abs_of_nonneg (byn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:¬rt = 0⊢ 0 ≤ 1 + |r|neg.refine_1n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:¬rt = 0⊢ 0 < (1 + |r|) ^ rt positivityAll goals completed! 🐙neg.refine_1n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:¬rt = 0⊢ 0 < (1 + |r|) ^ rt)]neg.refine_1n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:¬rt = 0⊢ 0 < (1 + |r|) ^ rt
      positivityAll goals completed! 🐙
    simp only [norm_pow, Real.norm_eq_abs, Nat.succ_eq_add_one]neg.refine_2n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:¬rt = 0⊢ |1 + |r|| ^ rt ≤ |1 + ‖(slice i).symm (r, x)‖| ^ rt
    refine pow_le_pow_left₀ (byn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:¬rt = 0⊢ 0 ≤ |1 + |r|| positivityAll goals completed! 🐙) ?_ rt
    rw [abs_of_nonneg (byn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:¬rt = 0⊢ 0 ≤ 1 + |r|neg.refine_2n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:¬rt = 0⊢ 1 + |r| ≤ |1 + ‖(slice i).symm (r, x)‖| positivityAll goals completed! 🐙neg.refine_2n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:¬rt = 0⊢ 1 + |r| ≤ |1 + ‖(slice i).symm (r, x)‖|)]neg.refine_2n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:¬rt = 0⊢ 1 + |r| ≤ |1 + ‖(slice i).symm (r, x)‖|
    conv_rhs => rw [abs_of_nonneg (byn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:¬rt = 0⊢ 0 ≤ 1 + ‖(slice i).symm (r, x)‖ positivityAll goals completed! 🐙)]
    simp only [add_le_add_iff_left]neg.refine_2n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝh✝:¬rt = 0⊢ |r| ≤ ‖(slice i).symm (r, x)‖
    exact abs_right_le_norm_slice_symm i r xAll goals completed! 🐙
  refine (le_mul_inv_iff₀ ?_).mpr (le_trans ?_ (h0 ((slice i).symm (r, x))))refine_1n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ 0 < ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖refine_2n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ * ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖ ≤
  (1 + ‖(slice i).symm (r, x)‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖
  ·refine_1n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ 0 < ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖ simprefine_1n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ 0 < |1 + ‖(slice i).symm (r, x)‖| ^ rt
    by_cases hr : rt = 0posn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝhr:rt = 0⊢ 0 < |1 + ‖(slice i).symm (r, x)‖| ^ rtnegn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝhr:¬rt = 0⊢ 0 < |1 + ‖(slice i).symm (r, x)‖| ^ rt
    ·posn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝhr:rt = 0⊢ 0 < |1 + ‖(slice i).symm (r, x)‖| ^ rt subst rtposn:ℕm:ℕd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)x:Space dr:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ 0)⁻¹‖) volumeh0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (0 + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (0 + m) * ((Finset.Iic (0 + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (0 + m, n).1 * ((Finset.Iic (0 + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ 0 < |1 + ‖(slice i).symm (r, x)‖| ^ 0
      simpAll goals completed! 🐙
    positivityAll goals completed! 🐙
  trans (‖((slice i).symm (r, x))‖ ^ m * ‖(1 + ‖((slice i).symm (r, x))‖) ^ rt‖) *
    ‖iteratedFDeriv ℝ n ⇑η (((slice i).symm (r, x)))‖n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ * ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖ ≤
  ‖(slice i).symm (r, x)‖ ^ m * ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖ * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ ‖(slice i).symm (r, x)‖ ^ m * ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖ * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤
  (1 + ‖(slice i).symm (r, x)‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖
  ·n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ * ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖ ≤
  ‖(slice i).symm (r, x)‖ ^ m * ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖ * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ apply le_of_eqn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ * ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖ =
  ‖(slice i).symm (r, x)‖ ^ m * ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖ * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖
    simp [mul_assoc]n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ * |1 + ‖(slice i).symm (r, x)‖| ^ rt =
    |1 + ‖(slice i).symm (r, x)‖| ^ rt * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ∨
  (r = 0 ∧ x = 0) ∧ ¬m = 0
    leftn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ * |1 + ‖(slice i).symm (r, x)‖| ^ rt =
  |1 + ‖(slice i).symm (r, x)‖| ^ rt * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖
    ringAll goals completed! 🐙
  apply mul_le_mul_of_nonneg _ (byn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ rflAll goals completed! 🐙) (byn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ 0 ≤ ‖(slice i).symm (r, x)‖ ^ m * ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖ positivityAll goals completed! 🐙) (byn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ 0 ≤ ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ positivityAll goals completed! 🐙)
  trans (1 + ‖((slice i).symm (r, x))‖) ^ m * (1 + ‖((slice i).symm (r, x))‖) ^ rtn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ ‖(slice i).symm (r, x)‖ ^ m * ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖ ≤
  (1 + ‖(slice i).symm (r, x)‖) ^ m * (1 + ‖(slice i).symm (r, x)‖) ^ rtn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ (1 + ‖(slice i).symm (r, x)‖) ^ m * (1 + ‖(slice i).symm (r, x)‖) ^ rt ≤ (1 + ‖(slice i).symm (r, x)‖) ^ (rt + m)
  ·n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ ‖(slice i).symm (r, x)‖ ^ m * ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖ ≤
  (1 + ‖(slice i).symm (r, x)‖) ^ m * (1 + ‖(slice i).symm (r, x)‖) ^ rt refine mul_le_mul_of_nonneg ?_ ?_ (byn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ 0 ≤ ‖(slice i).symm (r, x)‖ ^ m positivityAll goals completed! 🐙) (byn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ 0 ≤ (1 + ‖(slice i).symm (r, x)‖) ^ rt positivityAll goals completed! 🐙)
    ·refine_1n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ ‖(slice i).symm (r, x)‖ ^ m ≤ (1 + ‖(slice i).symm (r, x)‖) ^ m apply pow_le_pow_left₀ (byn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ 0 ≤ ‖(slice i).symm (r, x)‖ positivityAll goals completed! 🐙) ?_ m
      simpAll goals completed! 🐙
    ·refine_2n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ ‖(1 + ‖(slice i).symm (r, x)‖) ^ rt‖ ≤ (1 + ‖(slice i).symm (r, x)‖) ^ rt simprefine_2n:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ |1 + ‖(slice i).symm (r, x)‖| ^ rt ≤ (1 + ‖(slice i).symm (r, x)‖) ^ rt
      rw [abs_of_nonneg (byn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ 0 ≤ 1 + ‖(slice i).symm (r, x)‖All goals completed! 🐙 positivityAll goals completed! 🐙All goals completed! 🐙)]All goals completed! 🐙
  apply le_of_eqn:ℕm:ℕd:ℕi:Fin d.succrt:ℕhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumeη:𝓢(Space d.succ, ℝ)h0:∀ (x : Space (d + 1)),
  (1 + ‖x‖) ^ (rt + m) * ‖iteratedFDeriv ℝ n (⇑η) x‖ ≤
    2 ^ (rt + m) * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝ := 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dr:ℝ⊢ (1 + ‖(slice i).symm (r, x)‖) ^ m * (1 + ‖(slice i).symm (r, x)‖) ^ rt = (1 + ‖(slice i).symm (r, x)‖) ^ (rt + m)
  ring_nfAll goals completed! 🐙

A.2. Integrability for of Schwartz maps on slices

              

@[fun_prop]
lemma schwartzMap_mul_iteratedFDeriv_integrable_slice_symm {d : ℕ} (n m : ℕ)
    (η : 𝓢(Space d.succ, ℝ))
    (x : Space d) (i : Fin d.succ) :
    Integrable (fun r => ‖(slice i).symm (r, x)‖ ^ m *
    ‖iteratedFDeriv ℝ n ⇑η ((slice i).symm (r, x))‖) volume := byd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Integrable (fun r => ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) volume
  obtain ⟨rt, hrt⟩ := schwartzMap_slice_bound (m := m) (n := n) (d := d) id:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Integrable (fun r => ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) volume
  obtain ⟨k, hrt, hbound, k_eq⟩ := hrt ηd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Integrable (fun r => ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) volume
  apply Integrable.mono' (g := fun t => k * ‖(1 + ‖t‖) ^ (rt)‖⁻¹)hgd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Integrable (fun t => k * ‖(1 + ‖t‖) ^ rt‖⁻¹) volumehfd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ AEStronglyMeasurable (fun r => ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) volumehd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∀ᵐ (a : ℝ), ‖‖(slice i).symm (a, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (a, x))‖‖ ≤ k * ‖(1 + ‖a‖) ^ rt‖⁻¹
  ·hgd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Integrable (fun t => k * ‖(1 + ‖t‖) ^ rt‖⁻¹) volume apply Integrable.const_mulhgd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Integrable (fun x => ‖(1 + ‖x‖) ^ rt‖⁻¹) volume
    simpa using hrtAll goals completed! 🐙
  ·hfd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ AEStronglyMeasurable (fun r => ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) volume apply Continuous.aestronglyMeasurablehfd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Continuous fun r => ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖
    apply Continuous.mulhf.hfd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Continuous fun r => ‖(slice i).symm (r, x)‖ ^ mhgd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Continuous fun r => ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖
    ·hf.hfd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Continuous fun r => ‖(slice i).symm (r, x)‖ ^ m fun_propAll goals completed! 🐙
    apply Continuous.normhgd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Continuous fun x_1 => iteratedFDeriv ℝ n (⇑η) ((slice i).symm (x_1, x))
    apply Continuous.comp'hg.hgd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Continuous (iteratedFDeriv ℝ n ⇑η)hg.hfd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Continuous fun x_1 => (slice i).symm (x_1, x)
    apply ContDiff.continuous_iteratedFDeriv (n := (n + 1 : ℕ))hg.hg.hmd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ↑n ≤ ↑(n + 1)hg.hg.hfd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ContDiff ℝ ↑(n + 1) ⇑ηhg.hfd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Continuous fun x_1 => (slice i).symm (x_1, x)
    exact Nat.cast_le.mpr (byd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ n ≤ n + 1 omegaAll goals completed! 🐙)
    exact η.smooth'.of_le ENat.LEInfty.outhg.hfd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Continuous fun x_1 => (slice i).symm (x_1, x)
    fun_propAll goals completed! 🐙
  ·hd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∀ᵐ (a : ℝ), ‖‖(slice i).symm (a, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (a, x))‖‖ ≤ k * ‖(1 + ‖a‖) ^ rt‖⁻¹ filter_upwards with thd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηt:ℝ⊢ ‖‖(slice i).symm (t, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (t, x))‖‖ ≤ k * ‖(1 + ‖t‖) ^ rt‖⁻¹
    rw [Real.norm_of_nonneg (byd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηt:ℝ⊢ 0 ≤ ‖(slice i).symm (t, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (t, x))‖hd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηt:ℝ⊢ ‖(slice i).symm (t, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (t, x))‖ ≤ k * ‖(1 + ‖t‖) ^ rt‖⁻¹ positivityAll goals completed! 🐙hd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηt:ℝ⊢ ‖(slice i).symm (t, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (t, x))‖ ≤ k * ‖(1 + ‖t‖) ^ rt‖⁻¹)]hd:ℕn:ℕm:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηt:ℝ⊢ ‖(slice i).symm (t, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (t, x))‖ ≤ k * ‖(1 + ‖t‖) ^ rt‖⁻¹
    exact hbound x tAll goals completed! 🐙lemma schwartzMap_integrable_slice_symm {d : ℕ} (i : Fin d.succ) (η : 𝓢(Space d.succ, ℝ))
    (x : Space d) : Integrable (fun r => η ((slice i).symm (r, x))) volume := byd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun r => η ((slice i).symm (r, x))) volume
  apply (schwartzMap_mul_iteratedFDeriv_integrable_slice_symm 0 0 η x i).congr'hgd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)x:Space d⊢ AEStronglyMeasurable (fun r => η ((slice i).symm (r, x))) volumehd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)x:Space d⊢ ∀ᵐ (a : ℝ),
  ‖‖(slice i).symm (a, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (a, x))‖‖ = ‖η ((slice i).symm (a, x))‖
  ·hgd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)x:Space d⊢ AEStronglyMeasurable (fun r => η ((slice i).symm (r, x))) volume fun_propAll goals completed! 🐙
  ·hd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)x:Space d⊢ ∀ᵐ (a : ℝ),
  ‖‖(slice i).symm (a, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (a, x))‖‖ = ‖η ((slice i).symm (a, x))‖ simpAll goals completed! 🐙

set_option maxSynthPendingDepth 10000 in
lemma schwartzMap_fderiv_integrable_slice_symm {d : ℕ} (η : 𝓢(Space d.succ, ℝ)) (x : Space d)
    (i : Fin d.succ) :
    Integrable (fun r => fderiv ℝ (fun x => η (((slice i).symm (r, x)))) x) volume := byd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volume
  apply Integrable.mono' (g := fun r =>
    ‖iteratedFDeriv ℝ 1 ⇑η ((slice i).symm (r, x))‖ * ‖(slice i).symm.toContinuousLinearMap.comp
          (ContinuousLinearMap.prod (0 : Space d →L[ℝ] ℝ) (ContinuousLinearMap.id ℝ (Space d)))‖)hgd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Integrable
  (fun r =>
    ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖)
  volumehfd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ AEStronglyMeasurable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volumehd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ ∀ᵐ (a : ℝ),
  ‖fderiv ℝ (fun x => η ((slice i).symm (a, x))) x‖ ≤
    ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (a, x))‖ *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖
  ·hgd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Integrable
  (fun r =>
    ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖)
  volume apply Integrable.mul_consthgd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Integrable (fun x_1 => ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (x_1, x))‖) volume
    simpa using (schwartzMap_mul_iteratedFDeriv_integrable_slice_symm 1 0 η x i)All goals completed! 🐙
  ·hfd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ AEStronglyMeasurable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volume apply Continuous.aestronglyMeasurablehfd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Continuous fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x
    refine Continuous.fderiv_one ?_ ?_hf.refine_1d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ ContDiff ℝ 1 (Function.uncurry fun r x => η ((slice i).symm (r, x)))hf.refine_2d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Continuous fun r => x
    ·hf.refine_1d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ ContDiff ℝ 1 (Function.uncurry fun r x => η ((slice i).symm (r, x))) exact (η.smooth'.of_le (byd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ 1 ≤ ↑⊤ simpAll goals completed! 🐙)).comp ((slice i).symm.contDiff)
    ·hf.refine_2d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Continuous fun r => x fun_propAll goals completed! 🐙
  ·hd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ ∀ᵐ (a : ℝ),
  ‖fderiv ℝ (fun x => η ((slice i).symm (a, x))) x‖ ≤
    ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (a, x))‖ *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ filter_upwards with rhd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝ⊢ ‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖
    have heq : fderiv ℝ (fun x => (slice i).symm (r, x)) x =
        (slice i).symm.toContinuousLinearMap.comp
          (ContinuousLinearMap.prod (0 : Space d →L[ℝ] ℝ)
            (ContinuousLinearMap.id ℝ (Space d))) := byd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volumehd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖
      ext x2d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝx2:Space di✝:Fin d.succ⊢ ((fderiv ℝ (fun x => (slice i).symm (r, x)) x) x2).val i✝ =
  ((↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))) x2).val i✝hd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖
      simp [fderiv_slice_symm_right_apply]hd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖hd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖
    rw [norm_iteratedFDeriv_one,hd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖fderiv ℝ (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖hd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (⇑η) ((slice i).symm (r, x)) ∘SL
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  ‖fderiv ℝ (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x)) fderiv_fun_comp _ _ (byd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ DifferentiableAt ℝ (fun x => (slice i).symm (r, x)) xhd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (⇑η) ((slice i).symm (r, x)) ∘SL
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  ‖fderiv ℝ (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x)) fun_propAll goals completed! 🐙hd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (⇑η) ((slice i).symm (r, x)) ∘SL
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  ‖fderiv ℝ (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x))), heqhd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (⇑η) ((slice i).symm (r, x)) ∘SL
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  ‖fderiv ℝ (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x))hd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (⇑η) ((slice i).symm (r, x)) ∘SL
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  ‖fderiv ℝ (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x))]hd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (⇑η) ((slice i).symm (r, x)) ∘SL
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  ‖fderiv ℝ (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x))
    ·hd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (⇑η) ((slice i).symm (r, x)) ∘SL
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  ‖fderiv ℝ (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ exact ContinuousLinearMap.opNorm_comp_le _ _All goals completed! 🐙
    ·d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝheq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x)) exact η.differentiableAtAll goals completed! 🐙@[fun_prop]
lemma schwartzMap_fderiv_left_integrable_slice_symm {d : ℕ} (η : 𝓢(Space d.succ, ℝ)) (x : Space d)
    (i : Fin d.succ) :
    Integrable (fun r => fderiv ℝ (fun r => η (((slice i).symm (r, x)))) r 1) volume := byd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Integrable (fun r => (fderiv ℝ (fun r => η ((slice i).symm (r, x))) r) 1) volume
  conv_lhs =>
    enter [r]d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝ| (fderiv ℝ (fun r => η ((slice i).symm (r, x))) r) 1
    simp only [Nat.succ_eq_add_one, one_mul]d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝ| (fderiv ℝ (fun r => η ((slice i).symm (r, x))) r) 1
    change fderiv ℝ (η ∘ fun r => ((slice i).symm (r, x))) r 1d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝ| (fderiv ℝ (⇑η ∘ fun r => (slice i).symm (r, x)) r) 1
    rw [fderiv_comp _ η.differentiableAt (byd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝ⊢ DifferentiableAt ℝ (fun r => (slice i).symm (r, x)) r fun_propAll goals completed! 🐙)]
    simp only [Nat.succ_eq_add_one, ContinuousLinearMap.coe_comp, Function.comp_apply,
      fderiv_slice_symm_left_apply]d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝ| (fderiv ℝ (⇑η) ((slice i).symm (r, x))) ((slice i).symm (1, 0))
    change (SchwartzMap.evalCLM ℝ (Space d.succ) ℝ (((slice i).symm (1, 0)))).comp
      (SchwartzMap.fderivCLM ℝ (Space d.succ) ℝ) η (((slice i).symm (r, x)))d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝ| ((SchwartzMap.evalCLM ℝ (Space d.succ) ℝ ((slice i).symm (1, 0)) ∘SL fderivCLM ℝ (Space d.succ) ℝ) η)
  ((slice i).symm (r, x))
    rw [← SchwartzMap.lineDerivOpCLM_eq]d:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succr:ℝ| ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (1, 0))) η) ((slice i).symm (r, x))
  exact schwartzMap_integrable_slice_symm _ _ _All goals completed! 🐙@[fun_prop]
lemma schwartzMap_iteratedFDeriv_norm_slice_symm_integrable {n} {d : ℕ} (η : 𝓢(Space d.succ, ℝ))
    (x : Space d) (i : Fin d.succ) :
    Integrable (fun r => ‖iteratedFDeriv ℝ n ⇑η (((slice i).symm (r, x)))‖) volume := byn:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Integrable (fun r => ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) volume
  simpa using schwartzMap_mul_iteratedFDeriv_integrable_slice_symm n 0 η x iAll goals completed! 🐙

@[fun_prop]
lemma schwartzMap_iteratedFDeriv_slice_symm_integrable {n} {d : ℕ} (η : 𝓢(Space d.succ, ℝ))
    (x : Space d) (i : Fin d.succ) :
    Integrable (fun r => iteratedFDeriv ℝ n ⇑η (((slice i).symm (r, x)))) volume := byn:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Integrable (fun r => iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) volume
  rw [← MeasureTheory.integrable_norm_iffn:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Integrable (fun a => ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (a, x))‖) volumen:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ AEStronglyMeasurable (fun r => iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) volumen:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Integrable (fun a => ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (a, x))‖) volumen:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ AEStronglyMeasurable (fun r => iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) volume]n:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Integrable (fun a => ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (a, x))‖) volumen:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ AEStronglyMeasurable (fun r => iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) volume
  ·n:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Integrable (fun a => ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (a, x))‖) volume fun_propAll goals completed! 🐙
  ·n:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ AEStronglyMeasurable (fun r => iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) volume apply Continuous.aestronglyMeasurablen:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Continuous fun r => iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))
    apply Continuous.comp'hgn:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Continuous (iteratedFDeriv ℝ n ⇑η)hfn:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Continuous fun x_1 => (slice i).symm (x_1, x)
    apply ContDiff.continuous_iteratedFDeriv (n := (n + 1 : ℕ))hg.hmn:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ ↑n ≤ ↑(n + 1)hg.hfn:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ ContDiff ℝ ↑(n + 1) ⇑ηhfn:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Continuous fun x_1 => (slice i).symm (x_1, x)
    exact Nat.cast_le.mpr (byn:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ n ≤ n + 1 omegaAll goals completed! 🐙)
    exact η.smooth'.of_le ENat.LEInfty.outhfn:ℕd:ℕη:𝓢(Space d.succ, ℝ)x:Space di:Fin d.succ⊢ Continuous fun x_1 => (slice i).symm (x_1, x)
    fun_propAll goals completed! 🐙

A.3. Continiuity of integrations of slices of Schwartz maps

lemma continuous_schwartzMap_slice_integral {d} (i : Fin d.succ) (η : 𝓢(Space d.succ, ℝ)) :
    Continuous (fun x : Space d => ∫ r : ℝ, η ((slice i).symm (r, x))) := byd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)⊢ Continuous fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))
  obtain ⟨rt, hrt⟩ := schwartzMap_slice_bound (m := 0) (n := 0) (d := d) id:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)rt:ℕhrt:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Continuous fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))
  obtain ⟨k, hrt, hbound, k_eq⟩ := hrt ηd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)rt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Continuous fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))
  apply MeasureTheory.continuous_of_dominated (bound := fun t => k * ‖(1 + ‖t‖) ^ (rt)‖⁻¹)hF_measd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)rt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∀ (x : Space d), AEStronglyMeasurable (fun a => η ((slice i).symm (a, x))) volumeh_boundd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)rt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∀ (x : Space d), ∀ᵐ (a : ℝ), ‖η ((slice i).symm (a, x))‖ ≤ k * ‖(1 + ‖a‖) ^ rt‖⁻¹bound_integrabled:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)rt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Integrable (fun t => k * ‖(1 + ‖t‖) ^ rt‖⁻¹) volumeh_contd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)rt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∀ᵐ (a : ℝ), Continuous fun x => η ((slice i).symm (a, x))
  ·hF_measd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)rt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∀ (x : Space d), AEStronglyMeasurable (fun a => η ((slice i).symm (a, x))) volume intro xhF_measd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)rt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space d⊢ AEStronglyMeasurable (fun a => η ((slice i).symm (a, x))) volume
    fun_propAll goals completed! 🐙
  ·h_boundd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)rt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∀ (x : Space d), ∀ᵐ (a : ℝ), ‖η ((slice i).symm (a, x))‖ ≤ k * ‖(1 + ‖a‖) ^ rt‖⁻¹ intro xh_boundd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)rt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space d⊢ ∀ᵐ (a : ℝ), ‖η ((slice i).symm (a, x))‖ ≤ k * ‖(1 + ‖a‖) ^ rt‖⁻¹
    filter_upwards with th_boundd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)rt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space dt:ℝ⊢ ‖η ((slice i).symm (t, x))‖ ≤ k * ‖(1 + ‖t‖) ^ rt‖⁻¹
    simpa using hbound x tAll goals completed! 🐙
  ·bound_integrabled:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)rt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Integrable (fun t => k * ‖(1 + ‖t‖) ^ rt‖⁻¹) volume apply Integrable.const_mulbound_integrabled:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)rt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Integrable (fun x => ‖(1 + ‖x‖) ^ rt‖⁻¹) volume
    simpa using hrtAll goals completed! 🐙
  ·h_contd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)rt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∀ᵐ (a : ℝ), Continuous fun x => η ((slice i).symm (a, x)) filter_upwards with th_contd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)rt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 0).1 * ((Finset.Iic (rt + 0, 0)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηt:ℝ⊢ Continuous fun x => η ((slice i).symm (t, x))
    fun_propAll goals completed! 🐙

A.4. Derivative of integrations of slices of Schwartz maps



lemma schwartzMap_slice_integral_hasFDerivAt {d : ℕ} (η : 𝓢(Space d.succ, ℝ)) (i : Fin d.succ)
    (x₀ : Space d) :
    HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x)))
      (∫ (r : ℝ), fderiv ℝ (fun x : Space d => η ((slice i).symm (r, x))) x₀) x₀ := byd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space d⊢ HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x)))
  (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀) x₀
  let F : Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))⊢ HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x)))
  (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀) x₀
  let F' : Space d → ℝ → Space d →L[ℝ] ℝ :=
    fun x₀ r => fderiv ℝ (fun x : Space d => η ((slice i).symm (r, x))) x₀d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀⊢ HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x)))
  (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀) x₀
  have hF : ∀ t, ∀ x, HasFDerivAt (F · t) (F' x t) x := by
    intro t xd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀t:ℝx:Space d⊢ HasFDerivAt (fun x => F x t) (F' x t) xd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) x⊢ HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x)))
  (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀) x₀
    dsimp only [F, F']d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀t:ℝx:Space d⊢ HasFDerivAt (fun x => η ((slice i).symm (t, x))) (fderiv ℝ (fun x => η ((slice i).symm (t, x))) x) xd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) x⊢ HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x)))
  (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀) x₀
    refine DifferentiableAt.hasFDerivAt ?_d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀t:ℝx:Space d⊢ DifferentiableAt ℝ (fun x => η ((slice i).symm (t, x))) xd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) x⊢ HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x)))
  (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀) x₀
    exact (η.differentiable.comp (byd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀t:ℝx:Space d⊢ Differentiable ℝ fun x => (slice i).symm (t, x)d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) x⊢ HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x)))
  (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀) x₀ fun_propAll goals completed! 🐙d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) x⊢ HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x)))
  (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀) x₀)).differentiableAtd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) x⊢ HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x)))
  (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀) x₀
  obtain ⟨rt, hrt⟩ := schwartzMap_slice_bound (m := 0) (n := 1) (d := d) id:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x)))
  (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀) x₀
  obtain ⟨k, hrt, hbound, k_eq⟩ := hrt ηd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x)))
  (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀) x₀
  show HasFDerivAt (fun x => ∫ (a : ℝ), F x a) (∫ (a : ℝ), F' x₀ a) x₀d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ HasFDerivAt (fun x => ∫ (a : ℝ), F x a) (∫ (a : ℝ), F' x₀ a) x₀
  apply hasFDerivAt_integral_of_dominated_of_fderiv_le
    (bound := fun t => (k * ‖(slice i).symm.toContinuousLinearMap.comp
          (ContinuousLinearMap.prod (0 : Space d →L[ℝ] ℝ) (ContinuousLinearMap.id ℝ (Space d)))‖)
          * ‖(1 + ‖t‖) ^ (rt)‖⁻¹)hsd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ?m.479 ∈ nhds x₀hF_measd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∀ᶠ (x : Space d) in nhds x₀, AEStronglyMeasurable (F x) volumehF_intd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Integrable (F x₀) volumehF'_measd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ AEStronglyMeasurable (F' x₀) volumeh_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∀ᵐ (a : ℝ),
  ∀ x ∈ ?m.479,
    ‖F' x a‖ ≤
      k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖a‖) ^ rt‖⁻¹bound_integrabled:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Integrable
  (fun t =>
    k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖t‖) ^ rt‖⁻¹)
  volumeh_diffd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∀ᵐ (a : ℝ), ∀ x ∈ ?m.479, HasFDerivAt (fun x => F x a) (F' x a) xd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Set (Space d)
  ·hsd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ?m.479 ∈ nhds x₀ exact Filter.univ_mem' (hF (F x₀ 0))All goals completed! 🐙
  ·hF_measd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∀ᶠ (x : Space d) in nhds x₀, AEStronglyMeasurable (F x) volume filter_upwards with xhF_measd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηx:Space d⊢ AEStronglyMeasurable (F x) volume
    fun_propAll goals completed! 🐙
  ·hF_intd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Integrable (F x₀) volume simpa [F] using schwartzMap_integrable_slice_symm i η x₀All goals completed! 🐙
  ·hF'_measd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ AEStronglyMeasurable (F' x₀) volume simp [F']hF'_measd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ AEStronglyMeasurable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀) volume
    apply Continuous.aestronglyMeasurablehF'_measd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Continuous fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀
    refine Continuous.fderiv_one ?_ ?_hF'_meas.refine_1d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ContDiff ℝ 1 (Function.uncurry fun r x => η ((slice i).symm (r, x)))hF'_meas.refine_2d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Continuous fun r => x₀
    ·hF'_meas.refine_1d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ContDiff ℝ 1 (Function.uncurry fun r x => η ((slice i).symm (r, x))) exact (η.smooth'.of_le (byd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ 1 ≤ ↑⊤ simpAll goals completed! 🐙)).comp ((slice i).symm.contDiff)
    ·hF'_meas.refine_2d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Continuous fun r => x₀ fun_propAll goals completed! 🐙
  ·h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∀ᵐ (a : ℝ),
  ∀ x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a),
    ‖F' x a‖ ≤
      k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖a‖) ^ rt‖⁻¹ filter_upwards with rh_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝ⊢ ∀ x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a),
  ‖F' x r‖ ≤
    k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹
    intro x _h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)⊢ ‖F' x r‖ ≤
  k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹
    have hb : ‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
        ‖iteratedFDeriv ℝ 1 ⇑η ((slice i).symm (r, x))‖ *
        ‖(slice i).symm.toContinuousLinearMap.comp
          (ContinuousLinearMap.prod (0 : Space d →L[ℝ] ℝ)
            (ContinuousLinearMap.id ℝ (Space d)))‖ := byd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space d⊢ HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x)))
  (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀) x₀h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖F' x r‖ ≤
  k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹
      have heq : fderiv ℝ (fun x => (slice i).symm (r, x)) x =
          (slice i).symm.toContinuousLinearMap.comp
            (ContinuousLinearMap.prod (0 : Space d →L[ℝ] ℝ)
              (ContinuousLinearMap.id ℝ (Space d))) := byd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space d⊢ HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x)))
  (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀) x₀d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖F' x r‖ ≤
  k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹
        ext x2d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)x2:Space di✝:Fin d.succ⊢ ((fderiv ℝ (fun x => (slice i).symm (r, x)) x) x2).val i✝ =
  ((↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))) x2).val i✝d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖F' x r‖ ≤
  k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹
        simp [fderiv_slice_symm_right_apply]d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖F' x r‖ ≤
  k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖F' x r‖ ≤
  k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹
      rw [norm_iteratedFDeriv_one,d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖fderiv ℝ (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (⇑η) ((slice i).symm (r, x)) ∘SL
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  ‖fderiv ℝ (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x))h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖F' x r‖ ≤
  k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹ fderiv_fun_comp _ _ (byd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ DifferentiableAt ℝ (fun x => (slice i).symm (r, x)) xd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (⇑η) ((slice i).symm (r, x)) ∘SL
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  ‖fderiv ℝ (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x))h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖F' x r‖ ≤
  k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹ fun_propAll goals completed! 🐙d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (⇑η) ((slice i).symm (r, x)) ∘SL
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  ‖fderiv ℝ (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x))h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖F' x r‖ ≤
  k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹), heqd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (⇑η) ((slice i).symm (r, x)) ∘SL
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  ‖fderiv ℝ (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x))d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (⇑η) ((slice i).symm (r, x)) ∘SL
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  ‖fderiv ℝ (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x))h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖F' x r‖ ≤
  k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹]d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (⇑η) ((slice i).symm (r, x)) ∘SL
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  ‖fderiv ℝ (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x))h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖F' x r‖ ≤
  k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹
      ·d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ ‖fderiv ℝ (⇑η) ((slice i).symm (r, x)) ∘SL
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  ‖fderiv ℝ (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖F' x r‖ ≤
  k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹ exact ContinuousLinearMap.opNorm_comp_le _ _All goals completed! 🐙h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖F' x r‖ ≤
  k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹
      ·d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)heq:fderiv ℝ (fun x => (slice i).symm (r, x)) x =
  ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x))h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖F' x r‖ ≤
  k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹ exact η.differentiableAth_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖F' x r‖ ≤
  k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖F' x r‖ ≤
  k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹
    refine le_trans hb ?_h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖r‖) ^ rt‖⁻¹
    rw [mul_right_commh_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  k * ‖(1 + ‖r‖) ^ rt‖⁻¹ * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  k * ‖(1 + ‖r‖) ^ rt‖⁻¹ * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖]h_boundd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  k * ‖(1 + ‖r‖) ^ rt‖⁻¹ * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖
    gcongrhbcd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηr:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)hb:‖fderiv ℝ (fun x => η ((slice i).symm (r, x))) x‖ ≤
  ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖⊢ ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹
    simpa using hbound x rAll goals completed! 🐙
  ·bound_integrabled:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Integrable
  (fun t =>
    k * ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ * ‖(1 + ‖t‖) ^ rt‖⁻¹)
  volume apply Integrable.const_mulbound_integrabled:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Integrable (fun x => ‖(1 + ‖x‖) ^ rt‖⁻¹) volume
    simpa using hrtAll goals completed! 🐙
  ·h_diffd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∀ᵐ (a : ℝ),
  ∀ x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a),
    HasFDerivAt (fun x => F x a) (F' x a) x filter_upwards with th_diffd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηt:ℝ⊢ ∀ x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a),
  HasFDerivAt (fun x => F x t) (F' x t) x
    intro x _h_diffd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx₀:Space dF:Space d → ℝ → ℝ := fun x r => η ((slice i).symm (r, x))F':Space d → ℝ → Space d →L[ℝ] ℝ := fun x₀ r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀hF:∀ (t : ℝ) (x : Space d), HasFDerivAt (fun x => F x t) (F' x t) xrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ 0 * ‖iteratedFDeriv ℝ 1 (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + 0, 1).1 * ((Finset.Iic (rt + 0, 1)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηt:ℝx:Space da✝:x ∈ fun a => HasFDerivAtFilter (fun x => F x (F x₀ 0)) (F' a (F x₀ 0)) (nhds a ×ˢ pure a)⊢ HasFDerivAt (fun x => F x t) (F' x t) x
    exact hF t xAll goals completed! 🐙

A.5. Differentiability as a slices of Schwartz maps

lemma schwartzMap_slice_integral_differentiable {d : ℕ} (η : 𝓢(Space d.succ, ℝ))
    (i : Fin d.succ) :
    Differentiable ℝ (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) :=
  fun x => (schwartzMap_slice_integral_hasFDerivAt η i x).differentiableAt

A.6. Smoothness as slices of Schwartz maps



lemma schwartzMap_slice_integral_contDiff {d : ℕ} (n : ℕ) (η : 𝓢(Space d.succ, ℝ))
    (i : Fin d.succ) :
    ContDiff ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) := byd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succ⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))
  revert ηd:ℕn:ℕi:Fin d.succ⊢ ∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))
  induction n with
  | zero =>zerod:ℕi:Fin d.succ⊢ ∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑0 fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))
    intro ηzerod:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)⊢ ContDiff ℝ ↑0 fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))
    simp only [Nat.succ_eq_add_one, CharP.cast_eq_zero, contDiff_zero]zerod:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)⊢ Continuous fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))
    exact continuous_schwartzMap_slice_integral i ηAll goals completed! 🐙
  | succ n ih =>succd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))⊢ ∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑(n + 1) fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))
    intro ηsuccd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)⊢ ContDiff ℝ ↑(n + 1) fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))
    simp only [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one]succd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)⊢ ContDiff ℝ (↑n + 1) fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))
    rw [contDiff_succ_iff_hasFDerivAtsuccd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)⊢ ∃ f', ContDiff ℝ (↑n) f' ∧ ∀ (x : Space d), HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) (f' x) xsuccd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)⊢ ∃ f', ContDiff ℝ (↑n) f' ∧ ∀ (x : Space d), HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) (f' x) x]succd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)⊢ ∃ f', ContDiff ℝ (↑n) f' ∧ ∀ (x : Space d), HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) (f' x) x
    use fun x₀ => (∫ (r : ℝ), fderiv ℝ (fun x : Space d => η ((slice i).symm (r, x))) x₀)hd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)⊢ (ContDiff ℝ ↑n fun x₀ => ∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀) ∧
  ∀ (x : Space d),
    HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x)))
      ((fun x₀ => ∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀) x) x
    apply And.introh.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)⊢ ContDiff ℝ ↑n fun x₀ => ∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀h.rightd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)⊢ ∀ (x : Space d),
  HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x)))
    (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) x
    ·h.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)⊢ ContDiff ℝ ↑n fun x₀ => ∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x₀ rw [contDiff_clm_apply_iffh.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)⊢ ∀ (y : Space d), ContDiff ℝ ↑n fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yh.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)⊢ ∀ (y : Space d), ContDiff ℝ ↑n fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y]h.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)⊢ ∀ (y : Space d), ContDiff ℝ ↑n fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
      intro yh.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space d⊢ ContDiff ℝ ↑n fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
      have hl : (fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η (((slice i).symm (r, x)))) x) y) =
          fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η (((slice i).symm (r, x)))) x y) := byd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succ⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))h.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y⊢ ContDiff ℝ ↑n fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
        funext xd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dx:Space d⊢ (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y =
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yh.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y⊢ ContDiff ℝ ↑n fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
        simp only [Nat.succ_eq_add_one]d:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dx:Space d⊢ (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y =
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yh.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y⊢ ContDiff ℝ ↑n fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
        rw [ContinuousLinearMap.integral_applyd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dx:Space d⊢ ∫ (x_1 : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (x_1, x))) x) y =
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yφ_intd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dx:Space d⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volumeφ_intd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dx:Space d⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volumeh.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y⊢ ContDiff ℝ ↑n fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y]φ_intd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dx:Space d⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volumeh.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y⊢ ContDiff ℝ ↑n fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
        exact schwartzMap_fderiv_integrable_slice_symm η x ih.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y⊢ ContDiff ℝ ↑n fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yh.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y⊢ ContDiff ℝ ↑n fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
      rw [hlh.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yh.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y]h.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
      have hl2 : (fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η (((slice i).symm (r, x)))) x) y)=
          fun x => ∫ (r : ℝ), LineDeriv.lineDerivOpCLM ℝ _ ((slice i).symm (0, y)) η
            (((slice i).symm (r, x))) := byd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succ⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))h.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yhl2:(fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
        funext xd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space d⊢ ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y =
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))h.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yhl2:(fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
        congre_fd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space d⊢ (fun r => (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun r =>
  ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))h.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yhl2:(fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
        funext te_fd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ (fderiv ℝ (fun x => η ((slice i).symm (t, x))) x) y =
  ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (t, x))h.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yhl2:(fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
        simp only [Nat.succ_eq_add_one, LineDeriv.lineDerivOpCLM_apply]e_fd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ (fderiv ℝ (fun x => η ((slice i).symm (t, x))) x) y =
  (LineDeriv.lineDerivOp ((slice i).symm (0, y)) η) ((slice i).symm (t, x))h.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yhl2:(fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
        rw [fderiv_fun_compe_fd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ (fderiv ℝ (⇑η) ((slice i).symm (t, x)) ∘SL fderiv ℝ (fun x => (slice i).symm (t, x)) x) y =
  (LineDeriv.lineDerivOp ((slice i).symm (0, y)) η) ((slice i).symm (t, x))e_f.hgd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (t, x))e_f.hfd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ DifferentiableAt ℝ (fun x => (slice i).symm (t, x)) xe_fd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ (fderiv ℝ (⇑η) ((slice i).symm (t, x)) ∘SL fderiv ℝ (fun x => (slice i).symm (t, x)) x) y =
  (LineDeriv.lineDerivOp ((slice i).symm (0, y)) η) ((slice i).symm (t, x))e_f.hgd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (t, x))e_f.hfd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ DifferentiableAt ℝ (fun x => (slice i).symm (t, x)) xh.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yhl2:(fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y]e_fd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ (fderiv ℝ (⇑η) ((slice i).symm (t, x)) ∘SL fderiv ℝ (fun x => (slice i).symm (t, x)) x) y =
  (LineDeriv.lineDerivOp ((slice i).symm (0, y)) η) ((slice i).symm (t, x))e_f.hgd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (t, x))e_f.hfd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ DifferentiableAt ℝ (fun x => (slice i).symm (t, x)) xh.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yhl2:(fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
        simp only [ContinuousLinearMap.coe_comp, Function.comp_apply,
          fderiv_slice_symm_right_apply, Nat.succ_eq_add_one]e_fd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ (fderiv ℝ (⇑η) ((slice i).symm (t, x))) ((slice i).symm (0, y)) =
  (LineDeriv.lineDerivOp ((slice i).symm (0, y)) η) ((slice i).symm (t, x))e_f.hgd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (t, x))e_f.hfd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ DifferentiableAt ℝ (fun x => (slice i).symm (t, x)) xh.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yhl2:(fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
        rw [SchwartzMap.lineDerivOp_apply_eq_fderive_fd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ (fderiv ℝ (⇑η) ((slice i).symm (t, x))) ((slice i).symm (0, y)) =
  (fderiv ℝ (⇑η) ((slice i).symm (t, x))) ((slice i).symm (0, y))e_f.hgd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (t, x))e_f.hfd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ DifferentiableAt ℝ (fun x => (slice i).symm (t, x)) xe_f.hgd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (t, x))e_f.hfd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ DifferentiableAt ℝ (fun x => (slice i).symm (t, x)) xh.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yhl2:(fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y]e_f.hgd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (t, x))e_f.hfd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ DifferentiableAt ℝ (fun x => (slice i).symm (t, x)) xh.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yhl2:(fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
        ·e_f.hgd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yx:Space dt:ℝ⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (t, x))h.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yhl2:(fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y exact η.differentiableAtAll goals completed! 🐙h.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yhl2:(fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
        fun_proph.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yhl2:(fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yh.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yhl2:(fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y
      rw [hl2h.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yhl2:(fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))⊢ ContDiff ℝ ↑n fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))h.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yhl2:(fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))⊢ ContDiff ℝ ↑n fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))]h.leftd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)y:Space dhl:(fun x => (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) yhl2:(fun x => ∫ (r : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) y) = fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))⊢ ContDiff ℝ ↑n fun x =>
  ∫ (r : ℝ), ((LineDeriv.lineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) ((slice i).symm (0, y))) η) ((slice i).symm (r, x))
      apply ihAll goals completed! 🐙
    ·h.rightd:ℕi:Fin d.succn:ℕih:∀ (η : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))η:𝓢(Space d.succ, ℝ)⊢ ∀ (x : Space d),
  HasFDerivAt (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x)))
    (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) x exact fun x => schwartzMap_slice_integral_hasFDerivAt η i xAll goals completed! 🐙

A.7. Iterated derivatives of integrations of slices of Schwartz maps

              

lemma schwartzMap_slice_integral_iteratedFDeriv_apply {d : ℕ} (n : ℕ) (η : 𝓢(Space d.succ, ℝ))
    (i : Fin d.succ) :
    ∀ x, ∀ y, iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x y =
      ∫ (r : ℝ), (iteratedFDeriv ℝ n η ((slice i).symm (r, x)))
      (fun j => (slice i).symm (0, y j)) := byd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succ⊢ ∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)
  induction n with
  | zero =>zerod:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succ⊢ ∀ (x : Space d) (y : Fin 0 → Space d),
  (iteratedFDeriv ℝ 0 (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ 0 (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)
    simpAll goals completed! 🐙
  | succ n ih =>succd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)⊢ ∀ (x : Space d) (y : Fin (n + 1) → Space d),
  (iteratedFDeriv ℝ (n + 1) (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ (n + 1) (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)
    intro x ysuccd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space d⊢ (iteratedFDeriv ℝ (n + 1) (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
  ∫ (r : ℝ), (iteratedFDeriv ℝ (n + 1) (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)
    have hdiff : Differentiable ℝ (iteratedFDeriv ℝ n (⇑η : Space d.succ → ℝ)) := byd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succ⊢ ∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)succd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ (iteratedFDeriv ℝ (n + 1) (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
  ∫ (r : ℝ), (iteratedFDeriv ℝ (n + 1) (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)
      apply ContDiff.differentiable_iteratedFDeriv (n := (n + 1 : ℕ))hmd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space d⊢ ↑n < ↑(n + 1)hfd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space d⊢ ContDiff ℝ ↑(n + 1) ⇑ηsuccd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ (iteratedFDeriv ℝ (n + 1) (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
  ∫ (r : ℝ), (iteratedFDeriv ℝ (n + 1) (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)
      ·hmd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space d⊢ ↑n < ↑(n + 1)succd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ (iteratedFDeriv ℝ (n + 1) (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
  ∫ (r : ℝ), (iteratedFDeriv ℝ (n + 1) (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j) exact Nat.cast_lt.mpr (byd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space d⊢ n < n + 1succd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ (iteratedFDeriv ℝ (n + 1) (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
  ∫ (r : ℝ), (iteratedFDeriv ℝ (n + 1) (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j) omegaAll goals completed! 🐙succd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ (iteratedFDeriv ℝ (n + 1) (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
  ∫ (r : ℝ), (iteratedFDeriv ℝ (n + 1) (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j))
      ·hfd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space d⊢ ContDiff ℝ ↑(n + 1) ⇑ηsuccd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ (iteratedFDeriv ℝ (n + 1) (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
  ∫ (r : ℝ), (iteratedFDeriv ℝ (n + 1) (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j) exact η.smooth'.of_le ENat.LEInfty.outsuccd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ (iteratedFDeriv ℝ (n + 1) (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
  ∫ (r : ℝ), (iteratedFDeriv ℝ (n + 1) (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)succd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ (iteratedFDeriv ℝ (n + 1) (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
  ∫ (r : ℝ), (iteratedFDeriv ℝ (n + 1) (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)
    calc _
      _ = ((fderiv ℝ (fun x => iteratedFDeriv ℝ n
          (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x (Fin.tail y)) x) (y 0)) := byd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ (iteratedFDeriv ℝ (n + 1) (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
  (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) (Fin.tail y)) x) (y 0)
        rw [iteratedFDeriv_succ_apply_leftd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ ((fderiv ℝ (iteratedFDeriv ℝ n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) (y 0)) (Fin.tail y) =
  (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) (Fin.tail y)) x) (y 0)d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ ((fderiv ℝ (iteratedFDeriv ℝ n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) (y 0)) (Fin.tail y) =
  (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) (Fin.tail y)) x) (y 0)]d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ ((fderiv ℝ (iteratedFDeriv ℝ n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) (y 0)) (Fin.tail y) =
  (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) (Fin.tail y)) x) (y 0)
        refine Eq.symm (fderiv_continuousMultilinear_apply_const_apply ?_ (Fin.tail y) (y 0))d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ DifferentiableAt ℝ (iteratedFDeriv ℝ n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x
        exact ((schwartzMap_slice_integral_contDiff (n + 1) η i).differentiable_iteratedFDeriv
          (Nat.cast_lt.mpr (byd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ n < n + 1 omegaAll goals completed! 🐙))).differentiableAt
      _ = (fderiv ℝ (fun x => ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x)))
          fun j => (slice i).symm (0, Fin.tail y j)) x) (y 0) := byd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) (Fin.tail y)) x) (y 0) =
  (fderiv ℝ
      (fun x => ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j))
      x)
    (y 0)
        conv_lhs =>
          enter [1, 2, x]d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x✝:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)x:Space d| (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) (Fin.tail y)
          rw [ih]d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x✝:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)x:Space d| ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)
      _ = (fderiv ℝ (fun x => ∫ (r : ℝ), (LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ)
          (fun j => (slice i).symm (0, Fin.tail y j)) η (((slice i).symm (r, x)))))) x (y 0) := byd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ (fderiv ℝ
      (fun x => ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j))
      x)
    (y 0) =
  (fderiv ℝ
      (fun x =>
        ∫ (r : ℝ),
          ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
            ((slice i).symm (r, x)))
      x)
    (y 0)
        congre_fd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ (fun x => ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)) =
  fun x =>
  ∫ (r : ℝ),
    ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
      ((slice i).symm (r, x))
        funext xe_fd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x✝:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)x:Space d⊢ (∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)) =
  ∫ (r : ℝ),
    ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
      ((slice i).symm (r, x))
        congre_f.e_fd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x✝:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)x:Space d⊢ (fun r => (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)) = fun r =>
  ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
    ((slice i).symm (r, x))
        funext te_f.e_fd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x✝:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)x:Space dt:ℝ⊢ ((iteratedFDeriv ℝ n (⇑η) ((slice i).symm (t, x))) fun j => (slice i).symm (0, Fin.tail y j)) =
  ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
    ((slice i).symm (t, x))
        erw [SchwartzMap.iteratedLineDerivOp_eq_iteratedFDerive_f.e_fd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x✝:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)x:Space dt:ℝ⊢ ((iteratedFDeriv ℝ n (⇑η) ((slice i).symm (t, x))) fun j => (slice i).symm (0, Fin.tail y j)) =
  (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (t, x))) fun j => (slice i).symm (0, Fin.tail y j)]All goals completed! 🐙
      _ = ∫ (r : ℝ), (fderiv ℝ (fun x => ((LineDeriv.iteratedLineDerivOpCLM ℝ _ fun j =>
          (slice i).symm (0, Fin.tail y j)) η)
          ((slice i).symm (r, x))) x) (y 0) := byd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ (fderiv ℝ
      (fun x =>
        ∫ (r : ℝ),
          ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
            ((slice i).symm (r, x)))
      x)
    (y 0) =
  ∫ (r : ℝ),
    (fderiv ℝ
        (fun x =>
          ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
            ((slice i).symm (r, x)))
        x)
      (y 0)
        rw [(schwartzMap_slice_integral_hasFDerivAt _ i x).fderivd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ (∫ (r : ℝ),
      fderiv ℝ
        (fun x =>
          ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
            ((slice i).symm (r, x)))
        x)
    (y 0) =
  ∫ (r : ℝ),
    (fderiv ℝ
        (fun x =>
          ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
            ((slice i).symm (r, x)))
        x)
      (y 0)d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ (∫ (r : ℝ),
      fderiv ℝ
        (fun x =>
          ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
            ((slice i).symm (r, x)))
        x)
    (y 0) =
  ∫ (r : ℝ),
    (fderiv ℝ
        (fun x =>
          ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
            ((slice i).symm (r, x)))
        x)
      (y 0)]d:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ (∫ (r : ℝ),
      fderiv ℝ
        (fun x =>
          ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
            ((slice i).symm (r, x)))
        x)
    (y 0) =
  ∫ (r : ℝ),
    (fderiv ℝ
        (fun x =>
          ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
            ((slice i).symm (r, x)))
        x)
      (y 0)
        rw [ContinuousLinearMap.integral_applyd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ ∫ (x_1 : ℝ),
    (fderiv ℝ
        (fun x =>
          ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
            ((slice i).symm (x_1, x)))
        x)
      (y 0) =
  ∫ (r : ℝ),
    (fderiv ℝ
        (fun x =>
          ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
            ((slice i).symm (r, x)))
        x)
      (y 0)φ_intd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ Integrable
  (fun r =>
    fderiv ℝ
      (fun x =>
        ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
          ((slice i).symm (r, x)))
      x)
  volumeφ_intd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ Integrable
  (fun r =>
    fderiv ℝ
      (fun x =>
        ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
          ((slice i).symm (r, x)))
      x)
  volume]φ_intd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ Integrable
  (fun r =>
    fderiv ℝ
      (fun x =>
        ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
          ((slice i).symm (r, x)))
      x)
  volume
        exact
          schwartzMap_fderiv_integrable_slice_symm
            ((LineDeriv.iteratedLineDerivOpCLM ℝ _ fun j => (slice i).symm (0, Fin.tail y j)) η) x iAll goals completed! 🐙
    congrsucc.calc.step.e_fd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)⊢ (fun r =>
    (fderiv ℝ
        (fun x =>
          ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
            ((slice i).symm (r, x)))
        x)
      (y 0)) =
  fun r => (iteratedFDeriv ℝ (n + 1) (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)
    funext rsucc.calc.step.e_fd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ (fderiv ℝ
      (fun x =>
        ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
          ((slice i).symm (r, x)))
      x)
    (y 0) =
  (iteratedFDeriv ℝ (n + 1) (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)
    calc _
        _ = (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))
            (fun j => (slice i).symm (0, Fin.tail y j)))) x) (y 0) := byd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ (fderiv ℝ
      (fun x =>
        ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
          ((slice i).symm (r, x)))
      x)
    (y 0) =
  (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)) x)
    (y 0)
          congre_fd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ (fun x =>
    ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
      ((slice i).symm (r, x))) =
  fun x => (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)
          funext xe_fd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x✝:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝx:Space d⊢ ((LineDeriv.iteratedLineDerivOpCLM ℝ 𝓢(Space d.succ, ℝ) fun j => (slice i).symm (0, Fin.tail y j)) η)
    ((slice i).symm (r, x)) =
  (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)
          erw [SchwartzMap.iteratedLineDerivOp_eq_iteratedFDerive_fd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x✝:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝx:Space d⊢ ((iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)) =
  (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)]All goals completed! 🐙
    rw [iteratedFDeriv_succ_apply_leftsucc.calc.step.e_f.calc.stepd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)) x)
    (y 0) =
  ((fderiv ℝ (iteratedFDeriv ℝ n ⇑η) ((slice i).symm (r, x))) ((slice i).symm (0, y 0)))
    (Fin.tail fun j => (slice i).symm (0, y j))succ.calc.step.e_f.calc.stepd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)) x)
    (y 0) =
  ((fderiv ℝ (iteratedFDeriv ℝ n ⇑η) ((slice i).symm (r, x))) ((slice i).symm (0, y 0)))
    (Fin.tail fun j => (slice i).symm (0, y j))]succ.calc.step.e_f.calc.stepd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)) x)
    (y 0) =
  ((fderiv ℝ (iteratedFDeriv ℝ n ⇑η) ((slice i).symm (r, x))) ((slice i).symm (0, y 0)))
    (Fin.tail fun j => (slice i).symm (0, y j))
    simp only [Nat.succ_eq_add_one]succ.calc.step.e_f.calc.stepd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)) x)
    (y 0) =
  ((fderiv ℝ (iteratedFDeriv ℝ n ⇑η) ((slice i).symm (r, x))) ((slice i).symm (0, y 0)))
    (Fin.tail fun j => (slice i).symm (0, y j))
    rw [← fderiv_continuousMultilinear_apply_const_apply,succ.calc.step.e_f.calc.stepd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)) x)
    (y 0) =
  (fderiv ℝ (fun y_1 => (iteratedFDeriv ℝ n (⇑η) y_1) (Fin.tail fun j => (slice i).symm (0, y j)))
      ((slice i).symm (r, x)))
    ((slice i).symm (0, y 0))succ.calc.step.e_f.calc.step.hcd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ DifferentiableAt ℝ (iteratedFDeriv ℝ n ⇑η) ((slice i).symm (r, x))succ.calc.step.e_f.calc.stepd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)) x)
    (y 0) =
  (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) (Fin.tail fun j => (slice i).symm (0, y j))) x)
    (y 0)succ.calc.step.e_f.calc.step.hfd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ DifferentiableAt ℝ (fun y_1 => (iteratedFDeriv ℝ n (⇑η) y_1) (Fin.tail fun j => (slice i).symm (0, y j)))
  ((slice i).symm (r, x))succ.calc.step.e_f.calc.step.hcd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ DifferentiableAt ℝ (iteratedFDeriv ℝ n ⇑η) ((slice i).symm (r, x)) ← fderiv_fun_slice_symm_right_applysucc.calc.step.e_f.calc.stepd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)) x)
    (y 0) =
  (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) (Fin.tail fun j => (slice i).symm (0, y j))) x)
    (y 0)succ.calc.step.e_f.calc.step.hfd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ DifferentiableAt ℝ (fun y_1 => (iteratedFDeriv ℝ n (⇑η) y_1) (Fin.tail fun j => (slice i).symm (0, y j)))
  ((slice i).symm (r, x))succ.calc.step.e_f.calc.step.hcd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ DifferentiableAt ℝ (iteratedFDeriv ℝ n ⇑η) ((slice i).symm (r, x))succ.calc.step.e_f.calc.stepd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)) x)
    (y 0) =
  (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) (Fin.tail fun j => (slice i).symm (0, y j))) x)
    (y 0)succ.calc.step.e_f.calc.step.hfd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ DifferentiableAt ℝ (fun y_1 => (iteratedFDeriv ℝ n (⇑η) y_1) (Fin.tail fun j => (slice i).symm (0, y j)))
  ((slice i).symm (r, x))succ.calc.step.e_f.calc.step.hcd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ DifferentiableAt ℝ (iteratedFDeriv ℝ n ⇑η) ((slice i).symm (r, x))]succ.calc.step.e_f.calc.stepd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, Fin.tail y j)) x)
    (y 0) =
  (fderiv ℝ (fun x => (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) (Fin.tail fun j => (slice i).symm (0, y j))) x)
    (y 0)succ.calc.step.e_f.calc.step.hfd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ DifferentiableAt ℝ (fun y_1 => (iteratedFDeriv ℝ n (⇑η) y_1) (Fin.tail fun j => (slice i).symm (0, y j)))
  ((slice i).symm (r, x))succ.calc.step.e_f.calc.step.hcd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ DifferentiableAt ℝ (iteratedFDeriv ℝ n ⇑η) ((slice i).symm (r, x))
    rflsucc.calc.step.e_f.calc.step.hfd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ DifferentiableAt ℝ (fun y_1 => (iteratedFDeriv ℝ n (⇑η) y_1) (Fin.tail fun j => (slice i).symm (0, y j)))
  ((slice i).symm (r, x))succ.calc.step.e_f.calc.step.hcd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ DifferentiableAt ℝ (iteratedFDeriv ℝ n ⇑η) ((slice i).symm (r, x))
    ·succ.calc.step.e_f.calc.step.hfd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ DifferentiableAt ℝ (fun y_1 => (iteratedFDeriv ℝ n (⇑η) y_1) (Fin.tail fun j => (slice i).symm (0, y j)))
  ((slice i).symm (r, x)) exact (hdiff.continuousMultilinear_apply_const
        (Fin.tail fun j => (slice i).symm (0, y j))).differentiableAtAll goals completed! 🐙
    ·succ.calc.step.e_f.calc.step.hcd:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succn:ℕih:∀ (x : Space d) (y : Fin n → Space d),
  (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
    ∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)x:Space dy:Fin (n + 1) → Space dhdiff:Differentiable ℝ (iteratedFDeriv ℝ n ⇑η)r:ℝ⊢ DifferentiableAt ℝ (iteratedFDeriv ℝ n ⇑η) ((slice i).symm (r, x)) exact hdiff.differentiableAtAll goals completed! 🐙

lemma schwartzMap_slice_integral_iteratedFDeriv {d : ℕ} (n : ℕ) (η : 𝓢(Space d.succ, ℝ))
    (i : Fin d.succ) (x : Space d) :
    iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x
    = (∫ (r : ℝ), iteratedFDeriv ℝ n η ((slice i).symm (r, x))).compContinuousLinearMap
      (fun _ => (slice i).symm.toContinuousLinearMap.comp
      (ContinuousLinearMap.prod (0 : Space d →L[ℝ] ℝ) (ContinuousLinearMap.id ℝ (Space d)))) := byd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space d⊢ iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x =
  (∫ (r : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))).compContinuousLinearMap fun x =>
    ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))
  ext yd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space dy:Fin n → Space d⊢ (iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) y =
  ((∫ (r : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))).compContinuousLinearMap fun x =>
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d)))
    y
  rw [schwartzMap_slice_integral_iteratedFDeriv_apply,d:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space dy:Fin n → Space d⊢ (∫ (r : ℝ), (iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) fun j => (slice i).symm (0, y j)) =
  ((∫ (r : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))).compContinuousLinearMap fun x =>
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d)))
    yd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space dy:Fin n → Space d⊢ ((∫ (x_1 : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (x_1, x))) fun j => (slice i).symm (0, y j)) =
  ((∫ (r : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))).compContinuousLinearMap fun x =>
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d)))
    yφ_intd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space dy:Fin n → Space d⊢ Integrable (fun r => iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) volume ← ContinuousMultilinearMap.integral_applyd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space dy:Fin n → Space d⊢ ((∫ (x_1 : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (x_1, x))) fun j => (slice i).symm (0, y j)) =
  ((∫ (r : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))).compContinuousLinearMap fun x =>
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d)))
    yφ_intd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space dy:Fin n → Space d⊢ Integrable (fun r => iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) volumed:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space dy:Fin n → Space d⊢ ((∫ (x_1 : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (x_1, x))) fun j => (slice i).symm (0, y j)) =
  ((∫ (r : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))).compContinuousLinearMap fun x =>
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d)))
    yφ_intd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space dy:Fin n → Space d⊢ Integrable (fun r => iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) volume]d:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space dy:Fin n → Space d⊢ ((∫ (x_1 : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (x_1, x))) fun j => (slice i).symm (0, y j)) =
  ((∫ (r : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))).compContinuousLinearMap fun x =>
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d)))
    yφ_intd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space dy:Fin n → Space d⊢ Integrable (fun r => iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) volume
  ·d:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space dy:Fin n → Space d⊢ ((∫ (x_1 : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (x_1, x))) fun j => (slice i).symm (0, y j)) =
  ((∫ (r : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))).compContinuousLinearMap fun x =>
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d)))
    y rflAll goals completed! 🐙
  ·φ_intd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space dy:Fin n → Space d⊢ Integrable (fun r => iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))) volume exact schwartzMap_iteratedFDeriv_slice_symm_integrable η x iAll goals completed! 🐙

lemma schwartzMap_slice_integral_iteratedFDeriv_norm_le {d : ℕ} (n : ℕ) (η : 𝓢(Space d.succ, ℝ))
    (i : Fin d.succ) (x : Space d) :
    ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
        (∫ (r : ℝ), ‖iteratedFDeriv ℝ n η ((slice i).symm (r, x))‖) *
        ‖(slice i).symm.toContinuousLinearMap.comp
          (ContinuousLinearMap.prod (0 : Space d →L[ℝ] ℝ)
          (ContinuousLinearMap.id ℝ (Space d)))‖ ^ n := byd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space d⊢ ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
  (∫ (r : ℝ), ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n
  rw [schwartzMap_slice_integral_iteratedFDerivd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space d⊢ ‖(∫ (r : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))).compContinuousLinearMap fun x =>
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  (∫ (r : ℝ), ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ nd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space d⊢ ‖(∫ (r : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))).compContinuousLinearMap fun x =>
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  (∫ (r : ℝ), ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n]d:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space d⊢ ‖(∫ (r : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))).compContinuousLinearMap fun x =>
      ↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  (∫ (r : ℝ), ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n
  apply le_trans (ContinuousMultilinearMap.norm_compContinuousLinearMap_le _ _)d:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space d⊢ ‖∫ (r : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ *
    ∏ i_1, ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ≤
  (∫ (r : ℝ), ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n
  simpd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space d⊢ ‖∫ (r : ℝ), iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n ≤
  (∫ (r : ℝ), ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n
  refine mul_le_mul ?_ (byd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space d⊢ ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n ≤
  ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n rflAll goals completed! 🐙) (byd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space d⊢ 0 ≤ ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n positivityAll goals completed! 🐙) (byd:ℕn:ℕη:𝓢(Space d.succ, ℝ)i:Fin d.succx:Space d⊢ 0 ≤ ∫ (r : ℝ), ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ positivityAll goals completed! 🐙)
  exact norm_integral_le_integral_norm fun a => iteratedFDeriv ℝ n ⇑η _All goals completed! 🐙

lemma schwartzMap_mul_pow_slice_integral_iteratedFDeriv_norm_le {d : ℕ} (n m : ℕ) (i : Fin d.succ) :
    ∃ rt, ∀ (η : 𝓢(Space d.succ, ℝ)),∀ (x : Space d),
    Integrable (fun x : ℝ => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
        ((∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
        ‖(slice i).symm.toContinuousLinearMap.comp
          (ContinuousLinearMap.prod (0 : Space d →L[ℝ] ℝ)
          (ContinuousLinearMap.id ℝ (Space d)))‖ ^ n)
        * (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup
          fun m => SchwartzMap.seminorm ℝ m.1 m.2) η) := byd:ℕn:ℕm:ℕi:Fin d.succ⊢ ∃ rt,
  ∀ (η : 𝓢(Space d.succ, ℝ)) (x : Space d),
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
        (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
            ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
          (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)
  obtain ⟨rt, hrt⟩ := schwartzMap_slice_bound (m := m) (n := n) (d := d) id:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∃ rt,
  ∀ (η : 𝓢(Space d.succ, ℝ)) (x : Space d),
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
        (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
            ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
          (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)
  use rthd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ∀ (η : 𝓢(Space d.succ, ℝ)) (x : Space d),
  Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
      (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
          ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
        (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)
  intro η xhd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
  ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
    (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
        ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
      (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)
  obtain ⟨k, hrt, hbound, k_eq⟩ := hrt ηhd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
  ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
    (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
        ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
      (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)
  refine ⟨hrt, ?_⟩hd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k_eq:k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η⊢ ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
  (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
    (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)
  generalize hk : 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup
    fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k' at *hd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹k':ℝhk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k'k_eq:k = k'⊢ ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
  (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
    k'
  subst k_eqhd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
  (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
    k
  calc _
      _ ≤ ‖x‖ ^ m * ((∫ (r : ℝ), ‖iteratedFDeriv ℝ n η ((slice i).symm (r, x))‖) *
          ‖(slice i).symm.toContinuousLinearMap.comp
          ((0 : Space d →L[ℝ] ℝ).prod (ContinuousLinearMap.id ℝ (Space d)))‖ ^ n) := byd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
  ‖x‖ ^ m *
    ((∫ (r : ℝ), ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n)
        refine mul_le_mul_of_nonneg (byd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ ‖x‖ ^ m ≤ ‖x‖ ^ m rflAll goals completed! 🐙) ?_ (byd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ 0 ≤ ‖x‖ ^ m positivityAll goals completed! 🐙) (byd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ 0 ≤
  (∫ (r : ℝ), ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n positivityAll goals completed! 🐙)
        exact schwartzMap_slice_integral_iteratedFDeriv_norm_le n η i xAll goals completed! 🐙
      _ ≤ (∫ (r : ℝ), ‖x‖ ^ m * ‖iteratedFDeriv ℝ n η ((slice i).symm (r, x))‖) *
          ‖(slice i).symm.toContinuousLinearMap.comp
          ((0 : Space d →L[ℝ] ℝ).prod (ContinuousLinearMap.id ℝ (Space d)))‖ ^ n := byd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ ‖x‖ ^ m *
    ((∫ (r : ℝ), ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n) ≤
  (∫ (r : ℝ), ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n
        apply le_of_eqd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ ‖x‖ ^ m *
    ((∫ (r : ℝ), ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n) =
  (∫ (r : ℝ), ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n
        rw [← mul_assoc,d:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ (‖x‖ ^ m * ∫ (r : ℝ), ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n =
  (∫ (r : ℝ), ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ nAll goals completed! 🐙 MeasureTheory.integral_const_muld:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ (‖x‖ ^ m * ∫ (r : ℝ), ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n =
  (‖x‖ ^ m * ∫ (a : ℝ), ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (a, x))‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ nAll goals completed! 🐙]All goals completed! 🐙
      _ ≤ (∫ (r : ℝ), ‖((slice i).symm (r, x))‖ ^ m *
          ‖iteratedFDeriv ℝ n η (((slice i).symm (r, x)))‖) *
          ‖(slice i).symm.toContinuousLinearMap.comp
          ((0 : Space d →L[ℝ] ℝ).prod (ContinuousLinearMap.id ℝ (Space d)))‖ ^ n := byd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ (∫ (r : ℝ), ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n ≤
  (∫ (r : ℝ), ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n
        refine mul_le_mul_of_nonneg ?_ (byd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n ≤
  ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n rflAll goals completed! 🐙) (byd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ 0 ≤ ∫ (r : ℝ), ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ positivityAll goals completed! 🐙) (byd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ 0 ≤ ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n positivityAll goals completed! 🐙)
        refine integral_mono ?_ ?_ ?_refine_1d:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ Integrable (fun r => ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) volumerefine_2d:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ Integrable (fun r => ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) volumerefine_3d:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ (fun r => ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) ≤ fun r =>
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖
        ·refine_1d:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ Integrable (fun r => ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) volume apply Integrable.const_mulrefine_1d:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ Integrable (fun x_1 => ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (x_1, x))‖) volume
          fun_propAll goals completed! 🐙
        ·refine_2d:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ Integrable (fun r => ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) volume fun_propAll goals completed! 🐙
        ·refine_3d:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ (fun r => ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) ≤ fun r =>
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ intro trefine_3d:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = kt:ℝ⊢ (fun r => ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) t ≤
  (fun r => ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) t
          apply mul_le_mul_of_nonneg _ (byd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = kt:ℝ⊢ ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (t, x))‖ ≤ ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (t, x))‖ rflAll goals completed! 🐙) (byd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = kt:ℝ⊢ 0 ≤ ‖x‖ ^ m positivityAll goals completed! 🐙) (byd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = kt:ℝ⊢ 0 ≤ ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (t, x))‖ positivityAll goals completed! 🐙)
          refine pow_le_pow_left₀ (byd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = kt:ℝ⊢ 0 ≤ ‖x‖ positivityAll goals completed! 🐙) ?_ m
          simpAll goals completed! 🐙
      _ ≤ ((∫ (r : ℝ), k * ‖((1 + ‖r‖) ^ rt)⁻¹‖)) *
          ‖(slice i).symm.toContinuousLinearMap.comp
          ((0 : Space d →L[ℝ] ℝ).prod (ContinuousLinearMap.id ℝ (Space d)))‖ ^ n := byd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ (∫ (r : ℝ), ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n ≤
  (∫ (r : ℝ), k * ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n
        refine mul_le_mul_of_nonneg ?_ (byd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n ≤
  ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n rflAll goals completed! 🐙) (byd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ 0 ≤ ∫ (r : ℝ), ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ positivityAll goals completed! 🐙) (byd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ 0 ≤ ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n positivityAll goals completed! 🐙)
        refine integral_mono ?_ ?_ ?_refine_1d:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ Integrable (fun r => ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) volumerefine_2d:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ Integrable (fun r => k * ‖((1 + ‖r‖) ^ rt)⁻¹‖) volumerefine_3d:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ (fun r => ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) ≤ fun r =>
  k * ‖((1 + ‖r‖) ^ rt)⁻¹‖
        ·refine_1d:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ Integrable (fun r => ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) volume fun_propAll goals completed! 🐙
        ·refine_2d:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ Integrable (fun r => k * ‖((1 + ‖r‖) ^ rt)⁻¹‖) volume apply Integrable.const_mulrefine_2d:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume
          exact hrtAll goals completed! 🐙
        ·refine_3d:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ (fun r => ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) ≤ fun r =>
  k * ‖((1 + ‖r‖) ^ rt)⁻¹‖ intro trefine_3d:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = kt:ℝ⊢ (fun r => ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖) t ≤
  (fun r => k * ‖((1 + ‖r‖) ^ rt)⁻¹‖) t
          simpa using hbound x tAll goals completed! 🐙
  apply le_of_eqh.calc.stepd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ (∫ (r : ℝ), k * ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n =
  (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
    k
  rw [MeasureTheory.integral_const_mulh.calc.stepd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ (k * ∫ (a : ℝ), ‖((1 + ‖a‖) ^ rt)⁻¹‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n =
  (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
    kh.calc.stepd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ (k * ∫ (a : ℝ), ‖((1 + ‖a‖) ^ rt)⁻¹‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n =
  (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
    k]h.calc.stepd:ℕn:ℕm:ℕi:Fin d.succrt:ℕhrt✝:∀ (η : 𝓢(Space d.succ, ℝ)),
  ∃ k,
    Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
      (∀ (x : Space d) (r : ℝ),
          ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹) ∧
        k = 2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) ηη:𝓢(Space d.succ, ℝ)x:Space dk:ℝhrt:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:∀ (x : Space d) (r : ℝ),
  ‖(slice i).symm (r, x)‖ ^ m * ‖iteratedFDeriv ℝ n (⇑η) ((slice i).symm (r, x))‖ ≤ k * ‖(1 + ‖r‖) ^ rt‖⁻¹hk:2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η = k⊢ (k * ∫ (a : ℝ), ‖((1 + ‖a‖) ^ rt)⁻¹‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n =
  (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
    k
  ringAll goals completed! 🐙

A.8. The map integrating over one component of a Schwartz map

The continuous linear map taking a Schwartz map and integrating over the ith component, to give a Schwartz map of one dimension lower.


def sliceSchwartz {d : ℕ} (i : Fin d.succ) :
    𝓢(Space d.succ, ℝ) →L[ℝ] 𝓢(Space d, ℝ) := by𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succ⊢ 𝓢(Space d.succ, ℝ) →L[ℝ] 𝓢(Space d, ℝ)
  refine SchwartzMap.mkCLM (fun η x => ∫ (r : ℝ), η ((slice i).symm (r, x))) ?_ ?_ ?_ ?_refine_1𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succ⊢ ∀ (f g : 𝓢(Space d.succ, ℝ)) (x : Space d),
  ∫ (r : ℝ), (f + g) ((slice i).symm (r, x)) =
    (∫ (r : ℝ), f ((slice i).symm (r, x))) + ∫ (r : ℝ), g ((slice i).symm (r, x))refine_2𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succ⊢ ∀ (a : ℝ) (f : 𝓢(Space d.succ, ℝ)) (x : Space d),
  ∫ (r : ℝ), (a • f) ((slice i).symm (r, x)) = (RingHom.id ℝ) a • ∫ (r : ℝ), f ((slice i).symm (r, x))refine_3𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succ⊢ ∀ (f : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑⊤ fun x => ∫ (r : ℝ), f ((slice i).symm (r, x))refine_4𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succ⊢ ∀ (n : ℕ × ℕ),
  ∃ s C,
    0 ≤ C ∧
      ∀ (f : 𝓢(Space d.succ, ℝ)) (x : Space d),
        ‖x‖ ^ n.1 * ‖iteratedFDeriv ℝ n.2 (fun x => ∫ (r : ℝ), f ((slice i).symm (r, x))) x‖ ≤
          C * (s.sup (schwartzSeminormFamily ℝ (Space d.succ) ℝ)) f
  ·refine_1𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succ⊢ ∀ (f g : 𝓢(Space d.succ, ℝ)) (x : Space d),
  ∫ (r : ℝ), (f + g) ((slice i).symm (r, x)) =
    (∫ (r : ℝ), f ((slice i).symm (r, x))) + ∫ (r : ℝ), g ((slice i).symm (r, x)) intro η1 η2 xrefine_1𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succη1:𝓢(Space d.succ, ℝ)η2:𝓢(Space d.succ, ℝ)x:Space d⊢ ∫ (r : ℝ), (η1 + η2) ((slice i).symm (r, x)) =
  (∫ (r : ℝ), η1 ((slice i).symm (r, x))) + ∫ (r : ℝ), η2 ((slice i).symm (r, x))
    simp only [Nat.succ_eq_add_one, _root_.add_apply]refine_1𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succη1:𝓢(Space d.succ, ℝ)η2:𝓢(Space d.succ, ℝ)x:Space d⊢ ∫ (r : ℝ), η1 ((slice i).symm (r, x)) + η2 ((slice i).symm (r, x)) =
  (∫ (r : ℝ), η1 ((slice i).symm (r, x))) + ∫ (r : ℝ), η2 ((slice i).symm (r, x))
    rw [integral_addrefine_1𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succη1:𝓢(Space d.succ, ℝ)η2:𝓢(Space d.succ, ℝ)x:Space d⊢ (∫ (a : ℝ), η1 ((slice i).symm (a, x))) + ∫ (a : ℝ), η2 ((slice i).symm (a, x)) =
  (∫ (r : ℝ), η1 ((slice i).symm (r, x))) + ∫ (r : ℝ), η2 ((slice i).symm (r, x))refine_1.hf𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succη1:𝓢(Space d.succ, ℝ)η2:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun r => η1 ((slice i).symm (r, x))) volumerefine_1.hg𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succη1:𝓢(Space d.succ, ℝ)η2:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun r => η2 ((slice i).symm (r, x))) volumerefine_1.hf𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succη1:𝓢(Space d.succ, ℝ)η2:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun r => η1 ((slice i).symm (r, x))) volumerefine_1.hg𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succη1:𝓢(Space d.succ, ℝ)η2:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun r => η2 ((slice i).symm (r, x))) volume]refine_1.hf𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succη1:𝓢(Space d.succ, ℝ)η2:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun r => η1 ((slice i).symm (r, x))) volumerefine_1.hg𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succη1:𝓢(Space d.succ, ℝ)η2:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun r => η2 ((slice i).symm (r, x))) volume
    ·refine_1.hf𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succη1:𝓢(Space d.succ, ℝ)η2:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun r => η1 ((slice i).symm (r, x))) volume exact schwartzMap_integrable_slice_symm i η1 xAll goals completed! 🐙
    ·refine_1.hg𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succη1:𝓢(Space d.succ, ℝ)η2:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun r => η2 ((slice i).symm (r, x))) volume exact schwartzMap_integrable_slice_symm i η2 xAll goals completed! 🐙
  ·refine_2𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succ⊢ ∀ (a : ℝ) (f : 𝓢(Space d.succ, ℝ)) (x : Space d),
  ∫ (r : ℝ), (a • f) ((slice i).symm (r, x)) = (RingHom.id ℝ) a • ∫ (r : ℝ), f ((slice i).symm (r, x)) intro a η xrefine_2𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succa:ℝη:𝓢(Space d.succ, ℝ)x:Space d⊢ ∫ (r : ℝ), (a • η) ((slice i).symm (r, x)) = (RingHom.id ℝ) a • ∫ (r : ℝ), η ((slice i).symm (r, x))
    simp only [Nat.succ_eq_add_one, _root_.smul_apply, smul_eq_mul, RingHom.id_apply]refine_2𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succa:ℝη:𝓢(Space d.succ, ℝ)x:Space d⊢ ∫ (r : ℝ), a * η ((slice i).symm (r, x)) = a * ∫ (r : ℝ), η ((slice i).symm (r, x))
    rw [integral_const_mulrefine_2𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succa:ℝη:𝓢(Space d.succ, ℝ)x:Space d⊢ a * ∫ (a : ℝ), η ((slice i).symm (a, x)) = a * ∫ (r : ℝ), η ((slice i).symm (r, x))All goals completed! 🐙]All goals completed! 🐙
  ·refine_3𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succ⊢ ∀ (f : 𝓢(Space d.succ, ℝ)), ContDiff ℝ ↑⊤ fun x => ∫ (r : ℝ), f ((slice i).symm (r, x)) intro ηrefine_3𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)⊢ ContDiff ℝ ↑⊤ fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))
    simp only [Nat.succ_eq_add_one]refine_3𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)⊢ ContDiff ℝ ↑⊤ fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))
    refine contDiff_infty.mpr ?_refine_3𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)⊢ ∀ (n : ℕ), ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))
    intro nrefine_3𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)n:ℕ⊢ ContDiff ℝ ↑n fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))
    exact schwartzMap_slice_integral_contDiff n η iAll goals completed! 🐙
  ·refine_4𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succ⊢ ∀ (n : ℕ × ℕ),
  ∃ s C,
    0 ≤ C ∧
      ∀ (f : 𝓢(Space d.succ, ℝ)) (x : Space d),
        ‖x‖ ^ n.1 * ‖iteratedFDeriv ℝ n.2 (fun x => ∫ (r : ℝ), f ((slice i).symm (r, x))) x‖ ≤
          C * (s.sup (schwartzSeminormFamily ℝ (Space d.succ) ℝ)) f simprefine_4𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succ⊢ ∀ (a b : ℕ),
  ∃ s C,
    0 ≤ C ∧
      ∀ (f : 𝓢(Space (d + 1), ℝ)) (x : Space d),
        ‖x‖ ^ a * ‖iteratedFDeriv ℝ b (fun x => ∫ (r : ℝ), f ((slice i).symm (r, x))) x‖ ≤
          C * (s.sup (schwartzSeminormFamily ℝ (Space (d + 1)) ℝ)) f
    intro m nrefine_4𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succm:ℕn:ℕ⊢ ∃ s C,
  0 ≤ C ∧
    ∀ (f : 𝓢(Space (d + 1), ℝ)) (x : Space d),
      ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), f ((slice i).symm (r, x))) x‖ ≤
        C * (s.sup (schwartzSeminormFamily ℝ (Space (d + 1)) ℝ)) f
    obtain ⟨rt, hrt⟩ := schwartzMap_mul_pow_slice_integral_iteratedFDeriv_norm_le
      (d := d) (n := n) (m := m) irefine_4𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succm:ℕn:ℕrt:ℕhrt:∀ (η : 𝓢(Space d.succ, ℝ)) (x : Space d),
  Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
      (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
          ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
        (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)⊢ ∃ s C,
  0 ≤ C ∧
    ∀ (f : 𝓢(Space (d + 1), ℝ)) (x : Space d),
      ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), f ((slice i).symm (r, x))) x‖ ≤
        C * (s.sup (schwartzSeminormFamily ℝ (Space (d + 1)) ℝ)) f
    use (Finset.Iic (rt + m, n))h𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succm:ℕn:ℕrt:ℕhrt:∀ (η : 𝓢(Space d.succ, ℝ)) (x : Space d),
  Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
      (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
          ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
        (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)⊢ ∃ C,
  0 ≤ C ∧
    ∀ (f : 𝓢(Space (d + 1), ℝ)) (x : Space d),
      ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), f ((slice i).symm (r, x))) x‖ ≤
        C * ((Finset.Iic (rt + m, n)).sup (schwartzSeminormFamily ℝ (Space (d + 1)) ℝ)) f
    use 2 ^ (rt + m, n).1 * (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
        ‖(slice i).symm.toContinuousLinearMap.comp
        ((0 : Space d →L[ℝ] ℝ).prod (ContinuousLinearMap.id ℝ (Space d)))‖ ^ nh𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succm:ℕn:ℕrt:ℕhrt:∀ (η : 𝓢(Space d.succ, ℝ)) (x : Space d),
  Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
      (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
          ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
        (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)⊢ 0 ≤
    (2 ^ (rt + m, n).1 * ∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n ∧
  ∀ (f : 𝓢(Space (d + 1), ℝ)) (x : Space d),
    ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), f ((slice i).symm (r, x))) x‖ ≤
      (2 ^ (rt + m, n).1 * ∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
          ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
        ((Finset.Iic (rt + m, n)).sup (schwartzSeminormFamily ℝ (Space (d + 1)) ℝ)) f
    apply And.introh.left𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succm:ℕn:ℕrt:ℕhrt:∀ (η : 𝓢(Space d.succ, ℝ)) (x : Space d),
  Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
      (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
          ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
        (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)⊢ 0 ≤
  (2 ^ (rt + m, n).1 * ∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ nh.right𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succm:ℕn:ℕrt:ℕhrt:∀ (η : 𝓢(Space d.succ, ℝ)) (x : Space d),
  Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
      (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
          ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
        (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)⊢ ∀ (f : 𝓢(Space (d + 1), ℝ)) (x : Space d),
  ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), f ((slice i).symm (r, x))) x‖ ≤
    (2 ^ (rt + m, n).1 * ∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
        ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
      ((Finset.Iic (rt + m, n)).sup (schwartzSeminormFamily ℝ (Space (d + 1)) ℝ)) f
    ·h.left𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succm:ℕn:ℕrt:ℕhrt:∀ (η : 𝓢(Space d.succ, ℝ)) (x : Space d),
  Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
      (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
          ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
        (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)⊢ 0 ≤
  (2 ^ (rt + m, n).1 * ∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
    ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n positivityAll goals completed! 🐙
    intro η xh.right𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succm:ℕn:ℕrt:ℕhrt:∀ (η : 𝓢(Space d.succ, ℝ)) (x : Space d),
  Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
      (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
          ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
        (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)η:𝓢(Space (d + 1), ℝ)x:Space d⊢ ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
  (2 ^ (rt + m, n).1 * ∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
    ((Finset.Iic (rt + m, n)).sup (schwartzSeminormFamily ℝ (Space (d + 1)) ℝ)) η
    obtain ⟨hrt1, hbound⟩ := hrt η xh.right𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succm:ℕn:ℕrt:ℕhrt:∀ (η : 𝓢(Space d.succ, ℝ)) (x : Space d),
  Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
      (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
          ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
        (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)η:𝓢(Space (d + 1), ℝ)x:Space dhrt1:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
  (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
    (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)⊢ ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
  (2 ^ (rt + m, n).1 * ∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
    ((Finset.Iic (rt + m, n)).sup (schwartzSeminormFamily ℝ (Space (d + 1)) ℝ)) η
    apply le_trans hboundh.right𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succm:ℕn:ℕrt:ℕhrt:∀ (η : 𝓢(Space d.succ, ℝ)) (x : Space d),
  Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
      (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
          ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
        (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)η:𝓢(Space (d + 1), ℝ)x:Space dhrt1:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
  (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
    (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)⊢ (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
    (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η) ≤
  (2 ^ (rt + m, n).1 * ∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
    ((Finset.Iic (rt + m, n)).sup (schwartzSeminormFamily ℝ (Space (d + 1)) ℝ)) η
    apply le_of_eqh.right𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succm:ℕn:ℕrt:ℕhrt:∀ (η : 𝓢(Space d.succ, ℝ)) (x : Space d),
  Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
      (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
          ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
        (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)η:𝓢(Space (d + 1), ℝ)x:Space dhrt1:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
  (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
    (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)⊢ (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
    (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η) =
  (2 ^ (rt + m, n).1 * ∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
    ((Finset.Iic (rt + m, n)).sup (schwartzSeminormFamily ℝ (Space (d + 1)) ℝ)) η
    ring_nfh.right𝕜:TypeE:TypeF:TypeF':Typeinst✝⁵:RCLike 𝕜inst✝⁴:NormedAddCommGroup Einst✝³:NormedAddCommGroup Finst✝²:NormedAddCommGroup F'inst✝¹:NormedSpace ℝ Einst✝:NormedSpace ℝ Fd:ℕi:Fin d.succm:ℕn:ℕrt:ℕhrt:∀ (η : 𝓢(Space d.succ, ℝ)) (x : Space d),
  Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volume ∧
    ‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
      (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
          ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
        (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)η:𝓢(Space (d + 1), ℝ)x:Space dhrt1:Integrable (fun x => ‖((1 + ‖x‖) ^ rt)⁻¹‖) volumehbound:‖x‖ ^ m * ‖iteratedFDeriv ℝ n (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x‖ ≤
  (∫ (r : ℝ), ‖((1 + ‖r‖) ^ rt)⁻¹‖) *
      ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
    (2 ^ (rt + m, n).1 * ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η)⊢ (∫ (r : ℝ), ‖(1 + ‖r‖)⁻¹ ^ rt‖) *
          ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
        ((Finset.Iic (rt + m, n)).sup fun m => SchwartzMap.seminorm ℝ m.1 m.2) η *
      2 ^ rt *
    2 ^ m =
  (∫ (r : ℝ), ‖(1 + ‖r‖)⁻¹ ^ rt‖) *
          ‖↑(slice i).symm ∘SL ContinuousLinearMap.prod 0 (ContinuousLinearMap.id ℝ (Space d))‖ ^ n *
        ((Finset.Iic (rt + m, n)).sup (schwartzSeminormFamily ℝ (Space (d + 1)) ℝ)) η *
      2 ^ rt *
    2 ^ m
    rflAll goals completed! 🐙

lemma sliceSchwartz_apply {d : ℕ} (i : Fin d.succ) (η : 𝓢(Space d.succ, ℝ)) (x : Space d) :
    sliceSchwartz i η x = ∫ (r : ℝ), η ((slice i).symm (r, x)) := byd:ℕi:Fin d.succη:𝓢(Space d.succ, ℝ)x:Space d⊢ ((sliceSchwartz i) η) x = ∫ (r : ℝ), η ((slice i).symm (r, x))
  rflAll goals completed! 🐙

B. Constant slice distribution

Distributions on Space d.succ from distributions on Space d given a direction i. These distributions are constant on slices in the i direction..

def constantSliceDist {M : Type} [NormedAddCommGroup M] [NormedSpace ℝ M] {d : ℕ} (i : Fin d.succ) :
    ((Space d) →d[ℝ] M) →ₗ[ℝ] (Space d.succ) →d[ℝ] M where
  toFun f := f ∘L sliceSchwartz i
  map_add' f g := by𝕜:TypeE:TypeF:TypeF':Typeinst✝⁷:RCLike 𝕜inst✝⁶:NormedAddCommGroup Einst✝⁵:NormedAddCommGroup Finst✝⁴:NormedAddCommGroup F'inst✝³:NormedSpace ℝ Einst✝²:NormedSpace ℝ FM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mg:(Space d)→d[ℝ] M⊢ (f + g) ∘SL sliceSchwartz i = f ∘SL sliceSchwartz i + g ∘SL sliceSchwartz i
    ext η𝕜:TypeE:TypeF:TypeF':Typeinst✝⁷:RCLike 𝕜inst✝⁶:NormedAddCommGroup Einst✝⁵:NormedAddCommGroup Finst✝⁴:NormedAddCommGroup F'inst✝³:NormedSpace ℝ Einst✝²:NormedSpace ℝ FM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mg:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)⊢ ((f + g) ∘SL sliceSchwartz i) η = (f ∘SL sliceSchwartz i + g ∘SL sliceSchwartz i) η
    simpAll goals completed! 🐙
  map_smul' c f := by𝕜:TypeE:TypeF:TypeF':Typeinst✝⁷:RCLike 𝕜inst✝⁶:NormedAddCommGroup Einst✝⁵:NormedAddCommGroup Finst✝⁴:NormedAddCommGroup F'inst✝³:NormedSpace ℝ Einst✝²:NormedSpace ℝ FM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succc:ℝf:(Space d)→d[ℝ] M⊢ (c • f) ∘SL sliceSchwartz i = (RingHom.id ℝ) c • f ∘SL sliceSchwartz i
    ext η𝕜:TypeE:TypeF:TypeF':Typeinst✝⁷:RCLike 𝕜inst✝⁶:NormedAddCommGroup Einst✝⁵:NormedAddCommGroup Finst✝⁴:NormedAddCommGroup F'inst✝³:NormedSpace ℝ Einst✝²:NormedSpace ℝ FM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succc:ℝf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)⊢ ((c • f) ∘SL sliceSchwartz i) η = ((RingHom.id ℝ) c • f ∘SL sliceSchwartz i) η
    simpAll goals completed! 🐙

lemma constantSliceDist_apply {M : Type} [NormedAddCommGroup M] [NormedSpace ℝ M]
    {d : ℕ} (i : Fin d.succ) (f : (Space d) →d[ℝ] M) (η : 𝓢(Space d.succ, ℝ)) :
    constantSliceDist i f η = f (sliceSchwartz i η) := byM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)⊢ ((constantSliceDist i) f) η = f ((sliceSchwartz i) η)
  rflAll goals completed! 🐙

B.1. Derivative of constant slice distributions



lemma distDeriv_constantSliceDist_same {M : Type} [NormedAddCommGroup M] [NormedSpace ℝ M]
    {d : ℕ} (i : Fin d.succ) (f : (Space d) →d[ℝ] M) :
    distDeriv i (constantSliceDist i f) = 0 := byM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] M⊢ (distDeriv i) ((constantSliceDist i) f) = 0
  ext ηM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)⊢ ((distDeriv i) ((constantSliceDist i) f)) η = 0 η
  simp [constantSliceDist_apply, Space.distDeriv_apply, Distribution.fderivD_apply]M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)⊢ f ((sliceSchwartz i) ((SchwartzMap.evalCLM ℝ (Space (d + 1)) ℝ (basis i)) ((fderivCLM ℝ (Space (d + 1)) ℝ) η))) = 0
  trans f 0M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)⊢ f ((sliceSchwartz i) ((SchwartzMap.evalCLM ℝ (Space (d + 1)) ℝ (basis i)) ((fderivCLM ℝ (Space (d + 1)) ℝ) η))) = f 0M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)⊢ f 0 = 0; swapM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)⊢ f 0 = 0M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)⊢ f ((sliceSchwartz i) ((SchwartzMap.evalCLM ℝ (Space (d + 1)) ℝ (basis i)) ((fderivCLM ℝ (Space (d + 1)) ℝ) η))) = f 0
  ·M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)⊢ f 0 = 0 simpAll goals completed! 🐙
  congre_6M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)⊢ (sliceSchwartz i) ((SchwartzMap.evalCLM ℝ (Space (d + 1)) ℝ (basis i)) ((fderivCLM ℝ (Space (d + 1)) ℝ) η)) = 0
  ext xe_6M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ ((sliceSchwartz i) ((SchwartzMap.evalCLM ℝ (Space (d + 1)) ℝ (basis i)) ((fderivCLM ℝ (Space (d + 1)) ℝ) η))) x = 0 x
  simp [sliceSchwartz_apply]e_6M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ ∫ (r : ℝ), (fderiv ℝ (⇑η) ((slice i).symm (r, x))) (basis i) = 0
  calc _
    _ = ∫ r, fderiv ℝ η ((slice i).symm (r, x)) (basis i) := byM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ ∫ (r : ℝ), (fderiv ℝ (⇑η) ((slice i).symm (r, x))) (basis i) =
  ∫ (r : ℝ), (fderiv ℝ (⇑η) ((slice i).symm (r, x))) (basis i) rflAll goals completed! 🐙
    _ = ∫ r, fderiv ℝ (fun r => η ((slice i).symm (r, x))) r 1 := byM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ ∫ (r : ℝ), (fderiv ℝ (⇑η) ((slice i).symm (r, x))) (basis i) =
  ∫ (r : ℝ), (fderiv ℝ (fun r => η ((slice i).symm (r, x))) r) 1
        congre_fM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ (fun r => (fderiv ℝ (⇑η) ((slice i).symm (r, x))) (basis i)) = fun r =>
  (fderiv ℝ (fun r => η ((slice i).symm (r, x))) r) 1
        funext re_fM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space dr:ℝ⊢ (fderiv ℝ (⇑η) ((slice i).symm (r, x))) (basis i) = (fderiv ℝ (fun r => η ((slice i).symm (r, x))) r) 1
        rw [basis_self_eq_slice,e_fM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space dr:ℝ⊢ (fderiv ℝ (⇑η) ((slice i).symm (r, x))) ((slice i).symm (1, 0)) = (fderiv ℝ (fun r => η ((slice i).symm (r, x))) r) 1e_f.hfM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space dr:ℝ⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x)) fderiv_fun_slice_symm_left_applye_fM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space dr:ℝ⊢ (fderiv ℝ (⇑η) ((slice i).symm (r, x))) ((slice i).symm (1, 0)) =
  (fderiv ℝ (⇑η) ((slice i).symm (r, x))) ((slice i).symm (1, 0))e_f.hfM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space dr:ℝ⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x))e_f.hfM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space dr:ℝ⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x))]e_f.hfM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space dr:ℝ⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x))
        exact η.differentiableAtAll goals completed! 🐙
    _ = ∫ (r : ℝ), (fun r => 1) r * fderiv ℝ (fun r => η ((slice i).symm (r, x))) r 1 := byM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ ∫ (r : ℝ), (fderiv ℝ (fun r => η ((slice i).symm (r, x))) r) 1 =
  ∫ (r : ℝ), (fun r => 1) r * (fderiv ℝ (fun r => η ((slice i).symm (r, x))) r) 1 simpAll goals completed! 🐙
    _ = - ∫ (r : ℝ), fderiv ℝ (fun t => 1) r 1 * (fun r => η ((slice i).symm (r, x))) r := byM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ ∫ (r : ℝ), (fun r => 1) r * (fderiv ℝ (fun r => η ((slice i).symm (r, x))) r) 1 =
  -∫ (r : ℝ), (fderiv ℝ (fun t => 1) r) 1 * (fun r => η ((slice i).symm (r, x))) r
      rw [integral_mul_fderiv_eq_neg_fderiv_mul_of_integrableM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ -∫ (x_1 : ℝ), (fderiv ℝ (fun r => 1) x_1) 1 * η ((slice i).symm (x_1, x)) =
  -∫ (r : ℝ), (fderiv ℝ (fun t => 1) r) 1 * (fun r => η ((slice i).symm (r, x))) rhf'gM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun x_1 => (fderiv ℝ (fun r => 1) x_1) 1 * η ((slice i).symm (x_1, x))) volumehfg'M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun x_1 => 1 * (fderiv ℝ (fun r => η ((slice i).symm (r, x))) x_1) 1) volumehfgM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun x_1 => 1 * η ((slice i).symm (x_1, x))) volumehfM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ ∀ x_1 ∈ tsupport fun r => η ((slice i).symm (r, x)), DifferentiableAt ℝ (fun r => 1) x_1hgM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ ∀ x_1 ∈ tsupport fun r => 1, DifferentiableAt ℝ (fun r => η ((slice i).symm (r, x))) x_1hf'gM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun x_1 => (fderiv ℝ (fun r => 1) x_1) 1 * η ((slice i).symm (x_1, x))) volumehfg'M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun x_1 => 1 * (fderiv ℝ (fun r => η ((slice i).symm (r, x))) x_1) 1) volumehfgM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun x_1 => 1 * η ((slice i).symm (x_1, x))) volumehfM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ ∀ x_1 ∈ tsupport fun r => η ((slice i).symm (r, x)), DifferentiableAt ℝ (fun r => 1) x_1hgM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ ∀ x_1 ∈ tsupport fun r => 1, DifferentiableAt ℝ (fun r => η ((slice i).symm (r, x))) x_1]hf'gM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun x_1 => (fderiv ℝ (fun r => 1) x_1) 1 * η ((slice i).symm (x_1, x))) volumehfg'M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun x_1 => 1 * (fderiv ℝ (fun r => η ((slice i).symm (r, x))) x_1) 1) volumehfgM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun x_1 => 1 * η ((slice i).symm (x_1, x))) volumehfM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ ∀ x_1 ∈ tsupport fun r => η ((slice i).symm (r, x)), DifferentiableAt ℝ (fun r => 1) x_1hgM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ ∀ x_1 ∈ tsupport fun r => 1, DifferentiableAt ℝ (fun r => η ((slice i).symm (r, x))) x_1
      ·hf'gM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun x_1 => (fderiv ℝ (fun r => 1) x_1) 1 * η ((slice i).symm (x_1, x))) volume simpAll goals completed! 🐙
      ·hfg'M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun x_1 => 1 * (fderiv ℝ (fun r => η ((slice i).symm (r, x))) x_1) 1) volume simphfg'M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun x_1 => _root_.deriv (fun r => η ((slice i).symm (r, x))) x_1) volume
        change Integrable (fun r => fderiv ℝ (fun r => η ((slice i).symm (r, x))) r 1) volumehfg'M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun r => (fderiv ℝ (fun r => η ((slice i).symm (r, x))) r) 1) volume
        fun_propAll goals completed! 🐙
      ·hfgM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun x_1 => 1 * η ((slice i).symm (x_1, x))) volume simphfgM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ Integrable (fun x_1 => η ((slice i).symm (x_1, x))) volume
        exact schwartzMap_integrable_slice_symm i η xAll goals completed! 🐙
      ·hfM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ ∀ x_1 ∈ tsupport fun r => η ((slice i).symm (r, x)), DifferentiableAt ℝ (fun r => 1) x_1 fun_propAll goals completed! 🐙
      ·hgM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succf:(Space d)→d[ℝ] Mη:𝓢(Space d.succ, ℝ)x:Space d⊢ ∀ x_1 ∈ tsupport fun r => 1, DifferentiableAt ℝ (fun r => η ((slice i).symm (r, x))) x_1 fun_propAll goals completed! 🐙
  simpAll goals completed! 🐙



lemma distDeriv_constantSliceDist_succAbove {M : Type} [NormedAddCommGroup M] [NormedSpace ℝ M]
    {d : ℕ} (i : Fin d.succ) (j : Fin d) (f : (Space d) →d[ℝ] M) :
    distDeriv (i.succAbove j) (constantSliceDist i f) =
    constantSliceDist i (distDeriv j f) := byM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] M⊢ (distDeriv (i.succAbove j)) ((constantSliceDist i) f) = (constantSliceDist i) ((distDeriv j) f)
  ext ηM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)⊢ ((distDeriv (i.succAbove j)) ((constantSliceDist i) f)) η = ((constantSliceDist i) ((distDeriv j) f)) η
  simp [constantSliceDist_apply, Space.distDeriv_apply, Distribution.fderivD_apply]M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)⊢ f
    ((sliceSchwartz i)
      ((SchwartzMap.evalCLM ℝ (Space (d + 1)) ℝ (basis (i.succAbove j))) ((fderivCLM ℝ (Space (d + 1)) ℝ) η))) =
  f ((SchwartzMap.evalCLM ℝ (Space d) ℝ (basis j)) ((fderivCLM ℝ (Space d) ℝ) ((sliceSchwartz i) η)))
  congr 1e_6M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)⊢ (sliceSchwartz i)
    ((SchwartzMap.evalCLM ℝ (Space (d + 1)) ℝ (basis (i.succAbove j))) ((fderivCLM ℝ (Space (d + 1)) ℝ) η)) =
  (SchwartzMap.evalCLM ℝ (Space d) ℝ (basis j)) ((fderivCLM ℝ (Space d) ℝ) ((sliceSchwartz i) η))
  ext xe_6M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ ((sliceSchwartz i)
      ((SchwartzMap.evalCLM ℝ (Space (d + 1)) ℝ (basis (i.succAbove j))) ((fderivCLM ℝ (Space (d + 1)) ℝ) η)))
    x =
  ((SchwartzMap.evalCLM ℝ (Space d) ℝ (basis j)) ((fderivCLM ℝ (Space d) ℝ) ((sliceSchwartz i) η))) x
  simp [sliceSchwartz_apply]e_6M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ ∫ (r : ℝ), (fderiv ℝ (⇑η) ((slice i).symm (r, x))) (basis (i.succAbove j)) =
  (fderiv ℝ (⇑((sliceSchwartz i) η)) x) (basis j)
  change ∫ (r : ℝ), fderiv ℝ η _ _ = fderiv ℝ (fun x => ∫ (r : ℝ), η _) _ _e_6M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ ∫ (r : ℝ), (fderiv ℝ (⇑η) ((slice i).symm (r, x))) (basis (i.succAbove j)) =
  (fderiv ℝ (fun x => ∫ (r : ℝ), η ((slice i).symm (r, x))) x) (basis j)
  rw [(schwartzMap_slice_integral_hasFDerivAt η i x).fderiv,e_6M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ ∫ (r : ℝ), (fderiv ℝ (⇑η) ((slice i).symm (r, x))) (basis (i.succAbove j)) =
  (∫ (r : ℝ), fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) (basis j)e_6M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ ∫ (r : ℝ), (fderiv ℝ (⇑η) ((slice i).symm (r, x))) (basis (i.succAbove j)) =
  ∫ (x_1 : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (x_1, x))) x) (basis j)e_6.φ_intM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volume ContinuousLinearMap.integral_applye_6M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ ∫ (r : ℝ), (fderiv ℝ (⇑η) ((slice i).symm (r, x))) (basis (i.succAbove j)) =
  ∫ (x_1 : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (x_1, x))) x) (basis j)e_6.φ_intM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volumee_6M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ ∫ (r : ℝ), (fderiv ℝ (⇑η) ((slice i).symm (r, x))) (basis (i.succAbove j)) =
  ∫ (x_1 : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (x_1, x))) x) (basis j)e_6.φ_intM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volume]e_6M:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ ∫ (r : ℝ), (fderiv ℝ (⇑η) ((slice i).symm (r, x))) (basis (i.succAbove j)) =
  ∫ (x_1 : ℝ), (fderiv ℝ (fun x => η ((slice i).symm (x_1, x))) x) (basis j)e_6.φ_intM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volume
  congre_6.e_fM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ (fun r => (fderiv ℝ (⇑η) ((slice i).symm (r, x))) (basis (i.succAbove j))) = fun x_1 =>
  (fderiv ℝ (fun x => η ((slice i).symm (x_1, x))) x) (basis j)e_6.φ_intM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volume
  rw [basis_succAbove_eq_slicee_6.e_fM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ (fun r => (fderiv ℝ (⇑η) ((slice i).symm (r, x))) ((slice i).symm (0, basis j))) = fun x_1 =>
  (fderiv ℝ (fun x => η ((slice i).symm (x_1, x))) x) (basis j)e_6.φ_intM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volumee_6.e_fM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ (fun r => (fderiv ℝ (⇑η) ((slice i).symm (r, x))) ((slice i).symm (0, basis j))) = fun x_1 =>
  (fderiv ℝ (fun x => η ((slice i).symm (x_1, x))) x) (basis j)e_6.φ_intM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volume]e_6.e_fM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ (fun r => (fderiv ℝ (⇑η) ((slice i).symm (r, x))) ((slice i).symm (0, basis j))) = fun x_1 =>
  (fderiv ℝ (fun x => η ((slice i).symm (x_1, x))) x) (basis j)e_6.φ_intM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volume
  funext re_6.e_fM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space dr:ℝ⊢ (fderiv ℝ (⇑η) ((slice i).symm (r, x))) ((slice i).symm (0, basis j)) =
  (fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) (basis j)e_6.φ_intM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volume
  rw [fderiv_fun_slice_symm_right_applye_6.e_fM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space dr:ℝ⊢ (fderiv ℝ (⇑η) ((slice i).symm (r, x))) ((slice i).symm (0, basis j)) =
  (fderiv ℝ (⇑η) ((slice i).symm (r, x))) ((slice i).symm (0, basis j))e_6.e_f.hfM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space dr:ℝ⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x))e_6.φ_intM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volumee_6.e_f.hfM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space dr:ℝ⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x))e_6.φ_intM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volume]e_6.e_f.hfM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space dr:ℝ⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x))e_6.φ_intM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volume
  ·e_6.e_f.hfM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space dr:ℝ⊢ DifferentiableAt ℝ (⇑η) ((slice i).symm (r, x)) exact η.differentiableAtAll goals completed! 🐙
  ·e_6.φ_intM:Typeinst✝¹:NormedAddCommGroup Minst✝:NormedSpace ℝ Md:ℕi:Fin d.succj:Fin df:(Space d)→d[ℝ] Mη:𝓢(Space (d + 1), ℝ)x:Space d⊢ Integrable (fun r => fderiv ℝ (fun x => η ((slice i).symm (r, x))) x) volume exact schwartzMap_fderiv_integrable_slice_symm η x iAll goals completed! 🐙