set_option maxHeartbeats 700000 in -- 3.5× the default limit
/-- Bounding the norm of the iterated derivative of `B (f x) (g x)` within a set in terms of the
iterated derivatives of `f` and `g` when `B` is bilinear:
`‖D^n (x ↦ B (f x) (g x))‖ ≤ ‖B‖ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖` -/
theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear (B : E →L[𝕜] F →L[𝕜] G)
    {f : D → E} {g : D → F} {N : ℕ∞} {s : Set D} {x : D} (hf : ContDiffOn 𝕜 N f s)
    (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
    ‖iteratedFDerivWithin 𝕜 n (fun y => B (f y) (g y)) s x‖ ≤
      ‖B‖ * ∑ i in Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ := by
  /- We reduce the bound to the case where all spaces live in the same universe (in which we
    already have proved the result), by using linear isometries between the spaces and their `ULift`
    to a common universe. These linear isometries preserve the norm of the iterated derivative. -/
  let Du : Type max uD uE uF uG := ULift.{max uE uF uG, uD} D
  let Eu : Type max uD uE uF uG := ULift.{max uD uF uG, uE} E
  let Fu : Type max uD uE uF uG := ULift.{max uD uE uG, uF} F
  let Gu : Type max uD uE uF uG := ULift.{max uD uE uF, uG} G
  have isoD : Du ≃ₗᵢ[𝕜] D := LinearIsometryEquiv.ulift 𝕜 D
  have isoE : Eu ≃ₗᵢ[𝕜] E := LinearIsometryEquiv.ulift 𝕜 E
  have isoF : Fu ≃ₗᵢ[𝕜] F := LinearIsometryEquiv.ulift 𝕜 F
  have isoG : Gu ≃ₗᵢ[𝕜] G := LinearIsometryEquiv.ulift 𝕜 G
  -- lift `f` and `g` to versions `fu` and `gu` on the lifted spaces.
  set fu : Du → Eu := isoE.symm ∘ f ∘ isoD with hfu
  set gu : Du → Fu := isoF.symm ∘ g ∘ isoD with hgu
  -- lift the bilinear map `B` to a bilinear map `Bu` on the lifted spaces.
  set Bu₀ : Eu →L[𝕜] Fu →L[𝕜] G := ((B.comp (isoE : Eu →L[𝕜] E)).flip.comp (isoF : Fu →L[𝕜] F)).flip
    with hBu₀
  let Bu : Eu →L[𝕜] Fu →L[𝕜] Gu;
  exact ContinuousLinearMap.compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu)
    (ContinuousLinearMap.compL 𝕜 Fu G Gu (isoG.symm : G →L[𝕜] Gu)) Bu₀
  have hBu : Bu = ContinuousLinearMap.compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu)
      (ContinuousLinearMap.compL 𝕜 Fu G Gu (isoG.symm : G →L[𝕜] Gu)) Bu₀ := rfl
  have Bu_eq : (fun y => Bu (fu y) (gu y)) = isoG.symm ∘ (fun y => B (f y) (g y)) ∘ isoD := by
    ext1 y
    -- Porting note: the two blocks of `rw`s below were
    -- ```
    -- simp only [ContinuousLinearMap.compL_apply, Function.comp_apply,
    --   ContinuousLinearMap.coe_comp', LinearIsometryEquiv.coe_coe'',
    --   ContinuousLinearMap.flip_apply, LinearIsometryEquiv.apply_symm_apply]
    -- ```
    rw [hBu]
    iterate 2 rw [ContinuousLinearMap.compL_apply, ContinuousLinearMap.coe_comp',
      Function.comp_apply]
    rw [hBu₀]
    iterate 2 rw [ContinuousLinearMap.flip_apply, ContinuousLinearMap.coe_comp',
      Function.comp_apply]
    rw [hfu, Function.comp_apply, LinearIsometryEquiv.coe_coe'', LinearIsometryEquiv.coe_coe'',
      LinearIsometryEquiv.apply_symm_apply isoE, Function.comp_apply,
      hgu, LinearIsometryEquiv.coe_coe'', Function.comp_apply,
      LinearIsometryEquiv.apply_symm_apply isoF]
    simp only [Function.comp_apply]
  -- All norms are preserved by the lifting process.
  have Bu_le : ‖Bu‖ ≤ ‖B‖ := by
    refine' ContinuousLinearMap.op_norm_le_bound _ (norm_nonneg _) fun y => _
    refine' ContinuousLinearMap.op_norm_le_bound _ (by positivity) fun x => _
    simp only [ContinuousLinearMap.compL_apply, ContinuousLinearMap.coe_comp',
      Function.comp_apply, LinearIsometryEquiv.coe_coe'', ContinuousLinearMap.flip_apply,
      LinearIsometryEquiv.norm_map]
    rw [ContinuousLinearMap.coe_comp', Function.comp_apply, ContinuousLinearMap.compL_apply,
      ContinuousLinearMap.coe_comp', Function.comp_apply]
    iterate 2 rw [ContinuousLinearMap.flip_apply, ContinuousLinearMap.coe_comp',
      Function.comp_apply]
    simp only [LinearIsometryEquiv.coe_coe'', LinearIsometryEquiv.norm_map]
    calc
      ‖B (isoE y) (isoF x)‖ ≤ ‖B (isoE y)‖ * ‖isoF x‖ := ContinuousLinearMap.le_op_norm _ _
      _ ≤ ‖B‖ * ‖isoE y‖ * ‖isoF x‖ := by gcongr; apply ContinuousLinearMap.le_op_norm
      _ = ‖B‖ * ‖y‖ * ‖x‖ := by simp only [LinearIsometryEquiv.norm_map]
  let su := isoD ⁻¹' s
  have hsu : UniqueDiffOn 𝕜 su := isoD.toContinuousLinearEquiv.uniqueDiffOn_preimage_iff.2 hs
  let xu := isoD.symm x
  have hxu : xu ∈ su := by
    simpa only [Set.mem_preimage, LinearIsometryEquiv.apply_symm_apply] using hx
  have xu_x : isoD xu = x := by simp only [LinearIsometryEquiv.apply_symm_apply]
  have hfu : ContDiffOn 𝕜 n fu su :=
    isoE.symm.contDiff.comp_contDiffOn
      ((hf.of_le hn).comp_continuousLinearMap (isoD : Du →L[𝕜] D))
  have hgu : ContDiffOn 𝕜 n gu su :=
    isoF.symm.contDiff.comp_contDiffOn
      ((hg.of_le hn).comp_continuousLinearMap (isoD : Du →L[𝕜] D))
  have Nfu : ∀ i, ‖iteratedFDerivWithin 𝕜 i fu su xu‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖ := by
    intro i
    rw [LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_left _ _ hsu hxu]
    rw [LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_right _ _ hs, xu_x]
    rwa [← xu_x] at hx
  have Ngu : ∀ i, ‖iteratedFDerivWithin 𝕜 i gu su xu‖ = ‖iteratedFDerivWithin 𝕜 i g s x‖ := by
    intro i
    rw [LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_left _ _ hsu hxu]
    rw [LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_right _ _ hs, xu_x]
    rwa [← xu_x] at hx
  have NBu :
    ‖iteratedFDerivWithin 𝕜 n (fun y => Bu (fu y) (gu y)) su xu‖ =
      ‖iteratedFDerivWithin 𝕜 n (fun y => B (f y) (g y)) s x‖ := by
    rw [Bu_eq]
    rw [LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_left _ _ hsu hxu]
    rw [LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_right _ _ hs, xu_x]
    rwa [← xu_x] at hx
  -- state the bound for the lifted objects, and deduce the original bound from it.
  have : ‖iteratedFDerivWithin 𝕜 n (fun y => Bu (fu y) (gu y)) su xu‖ ≤
      ‖Bu‖ * ∑ i in Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i fu su xu‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) gu su xu‖ :=
    Bu.norm_iteratedFDerivWithin_le_of_bilinear_aux hfu hgu hsu hxu
  simp only [Nfu, Ngu, NBu] at this
  apply this.trans (mul_le_mul_of_nonneg_right Bu_le ?_)
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s✝ s₁ t u : Set E
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D
Eu : Type (max uD uE uF uG) := ULift.{max uD uF uG, uE} E
Fu : Type (max uD uE uF uG) := ULift.{max uD uE uG, uF} F
Gu : Type (max uD uE uF uG) := ULift.{max uD uE uF, uG} G
isoD : Du ≃ₗᵢ[𝕜] D
isoE : Eu ≃ₗᵢ[𝕜] E
isoF : Fu ≃ₗᵢ[𝕜] F
isoG : Gu ≃ₗᵢ[𝕜] G
fu : Du → Eu := ⇑(LinearIsometryEquiv.symm isoE) ∘ f ∘ ⇑isoD
hfu✝ : fu = ⇑(LinearIsometryEquiv.symm isoE) ∘ f ∘ ⇑isoD
gu : Du → Fu := ⇑(LinearIsometryEquiv.symm isoF) ∘ g ∘ ⇑isoD
hgu✝ : gu = ⇑(LinearIsometryEquiv.symm isoF) ∘ g ∘ ⇑isoD
Bu₀ : Eu →L[𝕜] Fu →L[𝕜] G :=
  flip
    (comp (flip (comp B ↑(ContinuousLinearEquiv.mk isoE.toLinearEquiv))) ↑(ContinuousLinearEquiv.mk isoF.toLinearEquiv))
hBu₀ :
  Bu₀ =
    flip
      (comp (flip (comp B ↑(ContinuousLinearEquiv.mk isoE.toLinearEquiv)))
        ↑(ContinuousLinearEquiv.mk isoF.toLinearEquiv))
Bu : Eu →L[𝕜] Fu →L[𝕜] Gu :=
  ((compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu))
      ((compL 𝕜 Fu G Gu) ↑(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm isoG).toLinearEquiv)))
    Bu₀
hBu :
  Bu =
    ((compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu))
        ((compL 𝕜 Fu G Gu) ↑(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm isoG).toLinearEquiv)))
      Bu₀
Bu_eq : (fun y => (Bu (fu y)) (gu y)) = ⇑(LinearIsometryEquiv.symm isoG) ∘ (fun y => (B (f y)) (g y)) ∘ ⇑isoD
Bu_le : ‖Bu‖ ≤ ‖B‖
su : Set Du := ⇑isoD ⁻¹' s
hsu : UniqueDiffOn 𝕜 su
xu : Du := (LinearIsometryEquiv.symm isoD) x
hxu : xu ∈ su
xu_x : isoD xu = x
hfu : ContDiffOn 𝕜 (↑n) fu su
hgu : ContDiffOn 𝕜 (↑n) gu su
Nfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu su xu‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖
Ngu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i gu su xu‖ = ‖iteratedFDerivWithin 𝕜 i g s x‖
NBu :
  ‖iteratedFDerivWithin 𝕜 n (fun y => (Bu (fu y)) (gu y)) su xu‖ =
    ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖
this :
  ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤
    ‖((compL 𝕜 (ULift.{max uD uF uG, uE} E) (ULift.{max uD uE uG, uF} F →L[𝕜] G)
              (ULift.{max uD uE uG, uF} F →L[𝕜] ULift.{max uD uE uF, uG} G))
            ((compL 𝕜 (ULift.{max uD uE uG, uF} F) G (ULift.{max uD uE uF, uG} G))
              ↑(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm isoG).toLinearEquiv)))
          (flip
            (comp (flip (comp B ↑(ContinuousLinearEquiv.mk isoE.toLinearEquiv)))
              ↑(ContinuousLinearEquiv.mk isoF.toLinearEquiv)))‖ *
      ∑ x_1 in Finset.range (n + 1),
        ↑(Nat.choose n x_1) * ‖iteratedFDerivWithin 𝕜 x_1 f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - x_1) g s x‖
⊢ 0 ≤
    ∑ x_1 in Finset.range (n + 1),
      ↑(Nat.choose n x_1) * ‖iteratedFDerivWithin 𝕜 x_1 f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - x_1) g s x‖

Next tactic:
exact Finset.sum_nonneg' fun i => by positivity