/-- If `f` is a local homeomorphism and the point `a` is in its target,
and if `f` is `n` times continuously differentiable at `f.symm a`,
and if the derivative at `f.symm a` is a continuous linear equivalence,
then `f.symm` is `n` times continuously differentiable at the point `a`.

This is one of the easy parts of the inverse function theorem: it assumes that we already have
an inverse function. -/
theorem PartialHomeomorph.contDiffAt_symm [CompleteSpace E] (f : PartialHomeomorph E F)
    {f₀' : E ≃L[𝕜] F} {a : F} (ha : a ∈ f.target)
    (hf₀' : HasFDerivAt f (f₀' : E →L[𝕜] F) (f.symm a)) (hf : ContDiffAt 𝕜 n f (f.symm a)) :
    ContDiffAt 𝕜 n f.symm a := by
  -- We prove this by induction on `n`
  induction' n using ENat.nat_induction with n IH Itop
  · rw [contDiffAt_zero]
    exact ⟨f.target, IsOpen.mem_nhds f.open_target ha, f.continuousOn_invFun⟩
  · obtain ⟨f', ⟨u, hu, hff'⟩, hf'⟩ := contDiffAt_succ_iff_hasFDerivAt.mp hf
    apply contDiffAt_succ_iff_hasFDerivAt.mpr
    -- For showing `n.succ` times continuous differentiability (the main inductive step), it
    -- suffices to produce the derivative and show that it is `n` times continuously differentiable
    have eq_f₀' : f' (f.symm a) = f₀' := (hff' (f.symm a) (mem_of_mem_nhds hu)).unique hf₀'
    -- This follows by a bootstrapping formula expressing the derivative as a function of `f` itself
    refine' ⟨inverse ∘ f' ∘ f.symm, _, _⟩
    · -- We first check that the derivative of `f` is that formula
      have h_nhds : { y : E | ∃ e : E ≃L[𝕜] F, ↑e = f' y } ∈ 𝓝 (f.symm a) := by
        have hf₀' := f₀'.nhds
        rw [← eq_f₀'] at hf₀'
        exact hf'.continuousAt.preimage_mem_nhds hf₀'
      obtain ⟨t, htu, ht, htf⟩ := mem_nhds_iff.mp (Filter.inter_mem hu h_nhds)
      use f.target ∩ f.symm ⁻¹' t
      refine' ⟨IsOpen.mem_nhds _ _, _⟩
      · exact f.isOpen_inter_preimage_symm ht
      · exact mem_inter ha (mem_preimage.mpr htf)
      intro x hx
      obtain ⟨hxu, e, he⟩ := htu hx.2
      have h_deriv : HasFDerivAt f (e : E →L[𝕜] F) (f.symm x) := by
        rw [he]
        exact hff' (f.symm x) hxu
      convert f.hasFDerivAt_symm hx.1 h_deriv
      simp [← he]
    · -- Then we check that the formula, being a composition of `ContDiff` pieces, is
      -- itself `ContDiff`
      have h_deriv₁ : ContDiffAt 𝕜 n inverse (f' (f.symm a)) := by
        rw [eq_f₀']
        exact contDiffAt_map_inverse _
      have h_deriv₂ : ContDiffAt 𝕜 n f.symm a := by
        refine' IH (hf.of_le _)
        norm_cast
        exact Nat.le_succ n
      

Proof state:
case hsuc.intro.intro.intro.intro.refine'_2
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
D : Type uD
inst✝¹⁰ : NormedAddCommGroup D
inst✝⁹ : NormedSpace 𝕜 D
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type uF
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
G : Type uG
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
X : Type u_2
inst✝² : NormedAddCommGroup X
inst✝¹ : NormedSpace 𝕜 X
s s₁ t u✝ : Set E
f✝ f₁ : E → F
g : F → G
x x₀ : E
c : F
b : E × F → G
m n✝ : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
inst✝ : CompleteSpace E
f : PartialHomeomorph E F
f₀' : E ≃L[𝕜] F
a : F
ha : a ∈ f.target
hf₀' : HasFDerivAt (↑f) (↑f₀') (↑(PartialHomeomorph.symm f) a)
n : ℕ
IH : ContDiffAt 𝕜 (↑n) (↑f) (↑(PartialHomeomorph.symm f) a) → ContDiffAt 𝕜 (↑n) (↑(PartialHomeomorph.symm f)) a
hf : ContDiffAt 𝕜 (↑(Nat.succ n)) (↑f) (↑(PartialHomeomorph.symm f) a)
f' : E → E →L[𝕜] F
hf' : ContDiffAt 𝕜 (↑n) f' (↑(PartialHomeomorph.symm f) a)
u : Set E
hu : u ∈ 𝓝 (↑(PartialHomeomorph.symm f) a)
hff' : ∀ x ∈ u, HasFDerivAt (↑f) (f' x) x
eq_f₀' : f' (↑(PartialHomeomorph.symm f) a) = ↑f₀'
h_deriv₁ : ContDiffAt 𝕜 (↑n) inverse (f' (↑(PartialHomeomorph.symm f) a))
h_deriv₂ : ContDiffAt 𝕜 (↑n) (↑(PartialHomeomorph.symm f)) a
⊢ ContDiffAt 𝕜 (↑n) (inverse ∘ f' ∘ ↑(PartialHomeomorph.symm f)) a

Next tactic:
exact (h_deriv₁.comp _ hf').comp _ h_deriv₂