/-- If the derivatives within a set of `g` at `f x` are bounded by `C`, and the `i`-th derivative
within a set of `f` at `x` is bounded by `D^i` for all `1 ≤ i ≤ n`, then the `n`-th derivative
of `g ∘ f` is bounded by `n! * C * D^n`.
This lemma proves this estimate assuming additionally that two of the spaces live in the same
universe, to make an induction possible. Use instead `norm_iteratedFDerivWithin_comp_le` that
removes this assumption. -/
theorem norm_iteratedFDerivWithin_comp_le_aux {Fu Gu : Type u} [NormedAddCommGroup Fu]
    [NormedSpace 𝕜 Fu] [NormedAddCommGroup Gu] [NormedSpace 𝕜 Gu] {g : Fu → Gu} {f : E → Fu} {n : ℕ}
    {s : Set E} {t : Set Fu} {x : E} (hg : ContDiffOn 𝕜 n g t) (hf : ContDiffOn 𝕜 n f s)
    (ht : UniqueDiffOn 𝕜 t) (hs : UniqueDiffOn 𝕜 s) (hst : MapsTo f s t) (hx : x ∈ s) {C : ℝ}
    {D : ℝ} (hC : ∀ i, i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C)
    (hD : ∀ i, 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i) :
    ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ n ! * C * D ^ n := by
  /- We argue by induction on `n`, using that `D^(n+1) (g ∘ f) = D^n (g ' ∘ f ⬝ f')`. The successive
    derivatives of `g' ∘ f` are controlled thanks to the inductive assumption, and those of `f'` are
    controlled by assumption.
    As composition of linear maps is a bilinear map, one may use
    `ContinuousLinearMap.norm_iteratedFDeriv_le_of_bilinear_of_le_one` to get from these a bound
    on `D^n (g ' ∘ f ⬝ f')`. -/
  induction' n using Nat.case_strong_induction_on with n IH generalizing Gu
  · simpa [norm_iteratedFDerivWithin_zero, Nat.factorial_zero, algebraMap.coe_one, one_mul,
      pow_zero, mul_one, comp_apply] using hC 0 le_rfl
  have M : (n : ℕ∞) < n.succ := Nat.cast_lt.2 n.lt_succ_self
  have Cnonneg : 0 ≤ C := (norm_nonneg _).trans (hC 0 bot_le)
  have Dnonneg : 0 ≤ D := by
    have : 1 ≤ n + 1 := by simp only [le_add_iff_nonneg_left, zero_le']
    simpa only [pow_one] using (norm_nonneg _).trans (hD 1 le_rfl this)
  -- use the inductive assumption to bound the derivatives of `g' ∘ f`.
  have I : ∀ i ∈ Finset.range (n + 1),
      ‖iteratedFDerivWithin 𝕜 i (fderivWithin 𝕜 g t ∘ f) s x‖ ≤ i ! * C * D ^ i
  · intro i hi
    simp only [Finset.mem_range_succ_iff] at hi
    apply IH i hi
    · apply hg.fderivWithin ht
      simp only [Nat.cast_succ]
      exact add_le_add_right (Nat.cast_le.2 hi) _
    · 

Proof state:
case I.hf
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
D✝ : Type uD
inst✝¹¹ : NormedAddCommGroup D✝
inst✝¹⁰ : NormedSpace 𝕜 D✝
E : Type uE
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type uF
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
G : Type uG
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
s✝ s₁ t✝ u : Set E
Fu : Type u
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
f : E → Fu
s : Set E
t : Set Fu
x : E
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hst : MapsTo f s t
hx : x ∈ s
C D : ℝ
n : ℕ
IH :
  ∀ m ≤ n,
    ∀ {Gu : Type u} [inst : NormedAddCommGroup Gu] [inst_1 : NormedSpace 𝕜 Gu] {g : Fu → Gu},
      ContDiffOn 𝕜 (↑m) g t →
        ContDiffOn 𝕜 (↑m) f s →
          (∀ i ≤ m, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C) →
            (∀ (i : ℕ), 1 ≤ i → i ≤ m → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i) →
              ‖iteratedFDerivWithin 𝕜 m (g ∘ f) s x‖ ≤ ↑m ! * C * D ^ m
Gu : Type u
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
g : Fu → Gu
hg : ContDiffOn 𝕜 (↑(Nat.succ n)) g t
hf : ContDiffOn 𝕜 (↑(Nat.succ n)) f s
hC : ∀ i ≤ Nat.succ n, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ Nat.succ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i
M : ↑n < ↑(Nat.succ n)
Cnonneg : 0 ≤ C
Dnonneg : 0 ≤ D
i : ℕ
hi : i ≤ n
⊢ ContDiffOn 𝕜 (↑i) f s

Next tactic:
apply hf.of_le (Nat.cast_le.2 (hi.trans n.le_succ))