/-- 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. This lemma is an auxiliary version
assuming all spaces live in the same universe, to enable an induction. Use instead
`ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear` that removes this assumption. -/
theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear_aux {Du Eu Fu Gu : Type u}
    [NormedAddCommGroup Du] [NormedSpace 𝕜 Du] [NormedAddCommGroup Eu] [NormedSpace 𝕜 Eu]
    [NormedAddCommGroup Fu] [NormedSpace 𝕜 Fu] [NormedAddCommGroup Gu] [NormedSpace 𝕜 Gu]
    (B : Eu →L[𝕜] Fu →L[𝕜] Gu) {f : Du → Eu} {g : Du → Fu} {n : ℕ} {s : Set Du} {x : Du}
    (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
    ‖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 argue by induction on `n`. The bound is trivial for `n = 0`. For `n + 1`, we write
    the `(n+1)`-th derivative as the `n`-th derivative of the derivative `B f g' + B f' g`,
    and apply the inductive assumption to each of those two terms. For this induction to make sense,
    the spaces of linear maps that appear in the induction should be in the same universe as the
    original spaces, which explains why we assume in the lemma that all spaces live in the same
    universe. -/
  induction' n with n IH generalizing Eu Fu Gu
  · simp only [Nat.zero_eq, norm_iteratedFDerivWithin_zero, zero_add, Finset.range_one,
      Finset.sum_singleton, Nat.choose_self, Nat.cast_one, one_mul, Nat.sub_zero, ← mul_assoc]
    apply B.le_op_norm₂
  · have In : (n : ℕ∞) + 1 ≤ n.succ := by simp only [Nat.cast_succ, le_refl]
    -- Porting note: the next line is a hack allowing Lean to find the operator norm instance.
    let norm := @ContinuousLinearMap.hasOpNorm _ _ Eu ((Du →L[𝕜] Fu) →L[𝕜] Du →L[𝕜] Gu) _ _ _ _ _ _
      (RingHom.id 𝕜)
    have I1 :
        ‖iteratedFDerivWithin 𝕜 n (fun y : Du => B.precompR Du (f y) (fderivWithin 𝕜 g s y)) s x‖ ≤
          ‖B‖ * ∑ i : ℕ in Finset.range (n + 1), n.choose i * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
            ‖iteratedFDerivWithin 𝕜 (n + 1 - i) g s x‖ := by
      calc
        ‖iteratedFDerivWithin 𝕜 n (fun y : Du => B.precompR Du (f y) (fderivWithin 𝕜 g s y)) s x‖ ≤
            ‖B.precompR Du‖ * ∑ i : ℕ in Finset.range (n + 1),
              n.choose i * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
                ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 g s) s x‖ :=
          IH _ (hf.of_le (Nat.cast_le.2 (Nat.le_succ n))) (hg.fderivWithin hs In)
        _ ≤ ‖B‖ * ∑ i : ℕ in Finset.range (n + 1), n.choose i * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
              ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 g s) s x‖ :=
          (mul_le_mul_of_nonneg_right (B.norm_precompR_le Du)
            (Finset.sum_nonneg' fun i => by positivity))
        _ = _ := by
          congr 1
          apply Finset.sum_congr rfl fun i hi => ?_
          rw [Nat.succ_sub (Nat.lt_succ_iff.1 (Finset.mem_range.1 hi)),
            ← norm_iteratedFDerivWithin_fderivWithin hs hx]
    -- Porting note: the next line is a hack allowing Lean to find the operator norm instance.
    let norm := @ContinuousLinearMap.hasOpNorm _ _ (Du →L[𝕜] Eu) (Fu →L[𝕜] Du →L[𝕜] Gu) _ _ _ _ _ _
      (RingHom.id 𝕜)
    have I2 :
        ‖iteratedFDerivWithin 𝕜 n (fun y : Du => B.precompL Du (fderivWithin 𝕜 f s y) (g y)) s x‖ ≤
        ‖B‖ * ∑ i : ℕ in Finset.range (n + 1), n.choose i * ‖iteratedFDerivWithin 𝕜 (i + 1) f s x‖ *
          ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ :=
      calc
        ‖iteratedFDerivWithin 𝕜 n (fun y : Du => B.precompL Du (fderivWithin 𝕜 f s y) (g y)) s x‖ ≤
            ‖B.precompL Du‖ * ∑ i : ℕ in Finset.range (n + 1),
              n.choose i * ‖iteratedFDerivWithin 𝕜 i (fderivWithin 𝕜 f s) s x‖ *
                ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ :=
          IH _ (hf.fderivWithin hs In) (hg.of_le (Nat.cast_le.2 (Nat.le_succ n)))
        _ ≤ ‖B‖ * ∑ i : ℕ in Finset.range (n + 1),
            n.choose i * ‖iteratedFDerivWithin 𝕜 i (fderivWithin 𝕜 f s) s x‖ *
              ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ :=
          (mul_le_mul_of_nonneg_right (B.norm_precompL_le Du)
            (Finset.sum_nonneg' fun i => by positivity))
        _ = _ := by
          congr 1
          apply Finset.sum_congr rfl fun i _ => ?_
          rw [← norm_iteratedFDerivWithin_fderivWithin hs hx]
    have J : iteratedFDerivWithin 𝕜 n
        (fun y : Du => fderivWithin 𝕜 (fun y : Du => B (f y) (g y)) s y) s x =
          iteratedFDerivWithin 𝕜 n (fun y => B.precompR Du (f y)
            (fderivWithin 𝕜 g s y) + B.precompL Du (fderivWithin 𝕜 f s y) (g y)) s x := by
      apply iteratedFDerivWithin_congr (fun y hy => ?_) hx
      have L : (1 : ℕ∞) ≤ n.succ := by
        simpa only [ENat.coe_one, Nat.one_le_cast] using Nat.succ_pos n
      exact B.fderivWithin_of_bilinear (hf.differentiableOn L y hy) (hg.differentiableOn L y hy)
        (hs y hy)
    rw [← norm_iteratedFDerivWithin_fderivWithin hs hx, J]
    have A : ContDiffOn 𝕜 n (fun y => B.precompR Du (f y) (fderivWithin 𝕜 g s y)) s :=
      (B.precompR Du).isBoundedBilinearMap.contDiff.comp_contDiff_on₂
        (hf.of_le (Nat.cast_le.2 (Nat.le_succ n))) (hg.fderivWithin hs In)
    have A' : ContDiffOn 𝕜 n (fun y => B.precompL Du (fderivWithin 𝕜 f s y) (g y)) s :=
      (B.precompL Du).isBoundedBilinearMap.contDiff.comp_contDiff_on₂ (hf.fderivWithin hs In)
        (hg.of_le (Nat.cast_le.2 (Nat.le_succ n)))
    rw [iteratedFDerivWithin_add_apply' A A' hs hx]
    apply (norm_add_le _ _).trans ((add_le_add I1 I2).trans (le_of_eq ?_))
    simp_rw [← mul_add, mul_assoc]
    congr 1
    

Proof state:
case e_a
𝕜 : Type u_1
inst✝¹⁶ : NontriviallyNormedField 𝕜
D : Type uD
inst✝¹⁵ : NormedAddCommGroup D
inst✝¹⁴ : NormedSpace 𝕜 D
E : Type uE
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
F : Type uF
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
G : Type uG
inst✝⁹ : NormedAddCommGroup G
inst✝⁸ : NormedSpace 𝕜 G
s✝ s₁ t u : Set E
Du : Type u
inst✝⁷ : NormedAddCommGroup Du
inst✝⁶ : NormedSpace 𝕜 Du
s : Set Du
x : Du
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
IH :
  ∀ {Eu Fu Gu : Type u} [inst : NormedAddCommGroup Eu] [inst_1 : NormedSpace 𝕜 Eu] [inst_2 : NormedAddCommGroup Fu]
    [inst_3 : NormedSpace 𝕜 Fu] [inst_4 : NormedAddCommGroup Gu] [inst_5 : NormedSpace 𝕜 Gu] (B : Eu →L[𝕜] Fu →L[𝕜] Gu)
    {f : Du → Eu} {g : Du → Fu},
    ContDiffOn 𝕜 (↑n) f s →
      ContDiffOn 𝕜 (↑n) g s →
        ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤
          ‖B‖ *
            ∑ i in Finset.range (n + 1),
              ↑(Nat.choose n i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
Eu Fu Gu : Type u
inst✝⁵ : NormedAddCommGroup Eu
inst✝⁴ : NormedSpace 𝕜 Eu
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
B : Eu →L[𝕜] Fu →L[𝕜] Gu
f : Du → Eu
g : Du → Fu
hf : ContDiffOn 𝕜 (↑(Nat.succ n)) f s
hg : ContDiffOn 𝕜 (↑(Nat.succ n)) g s
In : ↑n + 1 ≤ ↑(Nat.succ n)
norm✝ : Norm (Eu →L[𝕜] (Du →L[𝕜] Fu) →L[𝕜] Du →L[𝕜] Gu) := hasOpNorm
I1 :
  ‖iteratedFDerivWithin 𝕜 n (fun y => ((precompR Du B) (f y)) (fderivWithin 𝕜 g s y)) s x‖ ≤
    ‖B‖ *
      ∑ i in Finset.range (n + 1),
        ↑(Nat.choose n i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n + 1 - i) g s x‖
norm : Norm ((Du →L[𝕜] Eu) →L[𝕜] Fu →L[𝕜] Du →L[𝕜] Gu) := hasOpNorm
I2 :
  ‖iteratedFDerivWithin 𝕜 n (fun y => ((precompL Du B) (fderivWithin 𝕜 f s y)) (g y)) s x‖ ≤
    ‖B‖ *
      ∑ i in Finset.range (n + 1),
        ↑(Nat.choose n i) * ‖iteratedFDerivWithin 𝕜 (i + 1) f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
J :
  iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 (fun y => (B (f y)) (g y)) s y) s x =
    iteratedFDerivWithin 𝕜 n
      (fun y => ((precompR Du B) (f y)) (fderivWithin 𝕜 g s y) + ((precompL Du B) (fderivWithin 𝕜 f s y)) (g y)) s x
A : ContDiffOn 𝕜 (↑n) (fun y => ((precompR Du B) (f y)) (fderivWithin 𝕜 g s y)) s
A' : ContDiffOn 𝕜 (↑n) (fun y => ((precompL Du B) (fderivWithin 𝕜 f s y)) (g y)) s
⊢ ∑ x_1 in Finset.range (n + 1),
        ↑(Nat.choose n x_1) * (‖iteratedFDerivWithin 𝕜 x_1 f s x‖ * ‖iteratedFDerivWithin 𝕜 (n + 1 - x_1) g s x‖) +
      ∑ x_1 in Finset.range (n + 1),
        ↑(Nat.choose n x_1) * (‖iteratedFDerivWithin 𝕜 (x_1 + 1) f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - x_1) g s x‖) =
    ∑ x_1 in Finset.range (Nat.succ n + 1),
      ↑(Nat.choose (Nat.succ n) x_1) *
        (‖iteratedFDerivWithin 𝕜 x_1 f s x‖ * ‖iteratedFDerivWithin 𝕜 (Nat.succ n - x_1) g s x‖)

Next tactic:
exact (Finset.sum_choose_succ_mul
      (fun i j => ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 j g s x‖) n).symm