/-- A function is `C^(n + 1)` on a domain iff locally, it has a derivative which is `C^n`. -/
theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt {n : ℕ} :
    ContDiffWithinAt 𝕜 (n + 1 : ℕ) f s x ↔ ∃ u ∈ 𝓝[insert x s] x, ∃ f' : E → E →L[𝕜] F,
      (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffWithinAt 𝕜 n f' u x := by
  constructor
  · intro h
    rcases h n.succ le_rfl with ⟨u, hu, p, Hp⟩
    refine'
      ⟨u, hu, fun y => (continuousMultilinearCurryFin1 𝕜 E F) (p y 1), fun y hy =>
        Hp.hasFDerivWithinAt (WithTop.coe_le_coe.2 (Nat.le_add_left 1 n)) hy, _⟩
    intro m hm
    refine' ⟨u, _, fun y : E => (p y).shift, _⟩
    · -- Porting note: without the explicit argument Lean is not sure of the type.
      convert @self_mem_nhdsWithin _ _ x u
      have : x ∈ insert x s := by simp
      exact insert_eq_of_mem (mem_of_mem_nhdsWithin this hu)
    · rw [hasFTaylorSeriesUpToOn_succ_iff_right] at Hp
      exact Hp.2.2.of_le hm
  · rintro ⟨u, hu, f', f'_eq_deriv, Hf'⟩
    rw [contDiffWithinAt_nat]
    rcases Hf' n le_rfl with ⟨v, hv, p', Hp'⟩
    refine' ⟨v ∩ u, _, fun x => (p' x).unshift (f x), _⟩
    · apply Filter.inter_mem _ hu
      apply nhdsWithin_le_of_mem hu
      exact nhdsWithin_mono _ (subset_insert x u) hv
    · rw [hasFTaylorSeriesUpToOn_succ_iff_right]
      refine' ⟨fun y _ => rfl, fun y hy => _, _⟩
      · change
          HasFDerivWithinAt (fun z => (continuousMultilinearCurryFin0 𝕜 E F).symm (f z))
            (FormalMultilinearSeries.unshift (p' y) (f y) 1).curryLeft (v ∩ u) y
        -- Porting note: needed `erw` here.
        -- https://github.com/leanprover-community/mathlib4/issues/5164
        erw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff']
        convert (f'_eq_deriv y hy.2).mono (inter_subset_right v u)
        rw [← Hp'.zero_eq y hy.1]
        ext z
        change ((p' y 0) (init (@cons 0 (fun _ => E) z 0))) (@cons 0 (fun _ => E) z 0 (last 0)) =
          ((p' y 0) 0) z
        congr
        norm_num [eq_iff_true_of_subsingleton]
      · convert (Hp'.mono (inter_subset_left v u)).congr fun x hx => Hp'.zero_eq x hx.1 using 1
        · ext x y
          change p' x 0 (init (@snoc 0 (fun _ : Fin 1 => E) 0 y)) y = p' x 0 0 y
          rw [init_snoc]
        · ext x k v y
          

Proof state:
case h.e'_11.h.h.H.h
𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u✝ : Set E
f f₁ : E → F
g : F → G
x✝ x₀ : E
c : F
m n✝ : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
u : Set E
hu : u ∈ 𝓝[insert x✝ s] x✝
f' : E → E →L[𝕜] F
f'_eq_deriv : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
Hf' : ContDiffWithinAt 𝕜 (↑n) f' u x✝
v✝ : Set E
hv : v✝ ∈ 𝓝[insert x✝ u] x✝
p' : E → FormalMultilinearSeries 𝕜 E (E →L[𝕜] F)
Hp' : HasFTaylorSeriesUpToOn (↑n) f' p' v✝
x : E
k : ℕ
v : Fin k → E
y : E
⊢ ((FormalMultilinearSeries.shift (FormalMultilinearSeries.unshift (p' x) (f x)) k) v) y = ((p' x k) v) y

Next tactic:
change p' x k (init (@snoc k (fun _ : Fin k.succ => E) v y))
            (@snoc k (fun _ : Fin k.succ => E) v y (last k)) = p' x k v y