/-- If a function is linearly approximated by a continuous linear map with a (possibly nonlinear)
right inverse, then it is locally onto: a ball of an explicit radius is included in the image
of the map. -/
theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f f' s c)
    (f'symm : f'.NonlinearRightInverse) {ε : ℝ} {b : E} (ε0 : 0 ≤ ε) (hε : closedBall b ε ⊆ s) :
    SurjOn f (closedBall b ε) (closedBall (f b) (((f'symm.nnnorm : ℝ)⁻¹ - c) * ε)) := by
  intro y hy
  rcases le_or_lt (f'symm.nnnorm : ℝ)⁻¹ c with hc | hc
  · refine' ⟨b, by simp [ε0], _⟩
    have : dist y (f b) ≤ 0 :=
      (mem_closedBall.1 hy).trans (mul_nonpos_of_nonpos_of_nonneg (by linarith) ε0)
    simp only [dist_le_zero] at this
    rw [this]
  have If' : (0 : ℝ) < f'symm.nnnorm := by rw [← inv_pos]; exact (NNReal.coe_nonneg _).trans_lt hc
  have Icf' : (c : ℝ) * f'symm.nnnorm < 1 := by rwa [inv_eq_one_div, lt_div_iff If'] at hc
  have Jf' : (f'symm.nnnorm : ℝ) ≠ 0 := ne_of_gt If'
  have Jcf' : (1 : ℝ) - c * f'symm.nnnorm ≠ 0 := by apply ne_of_gt; linarith
  /- We have to show that `y` can be written as `f x` for some `x ∈ closedBall b ε`.
    The idea of the proof is to apply the Banach contraction principle to the map
    `g : x ↦ x + f'symm (y - f x)`, as a fixed point of this map satisfies `f x = y`.
    When `f'symm` is a genuine linear inverse, `g` is a contracting map. In our case, since `f'symm`
    is nonlinear, this map is not contracting (it is not even continuous), but still the proof of
    the contraction theorem holds: `uₙ = gⁿ b` is a Cauchy sequence, converging exponentially fast
    to the desired point `x`. Instead of appealing to general results, we check this by hand.

    The main point is that `f (u n)` becomes exponentially close to `y`, and therefore
    `dist (u (n+1)) (u n)` becomes exponentally small, making it possible to get an inductive
    bound on `dist (u n) b`, from which one checks that `u n` stays in the ball on which one has a
    control. Therefore, the bound can be checked at the next step, and so on inductively.
    -/
  set g := fun x => x + f'symm (y - f x) with hg
  set u := fun n : ℕ => g^[n] b with hu
  have usucc : ∀ n, u (n + 1) = g (u n) := by simp [hu, ← iterate_succ_apply' g _ b]
  -- First bound: if `f z` is close to `y`, then `g z` is close to `z` (i.e., almost a fixed point).
  

Proof state:
case inr
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
G : Type u_4
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
G' : Type u_5
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
ε✝ : ℝ
inst✝ : CompleteSpace E
f : E → F
s : Set E
c : ℝ≥0
f' : E →L[𝕜] F
hf : ApproximatesLinearOn f f' s c
f'symm : ContinuousLinearMap.NonlinearRightInverse f'
ε : ℝ
b : E
ε0 : 0 ≤ ε
hε : closedBall b ε ⊆ s
y : F
hy : y ∈ closedBall (f b) (((↑f'symm.nnnorm)⁻¹ - ↑c) * ε)
hc : ↑c < (↑f'symm.nnnorm)⁻¹
If' : 0 < ↑f'symm.nnnorm
Icf' : ↑c * ↑f'symm.nnnorm < 1
Jf' : ↑f'symm.nnnorm ≠ 0
Jcf' : 1 - ↑c * ↑f'symm.nnnorm ≠ 0
g : E → E := fun x => x + ContinuousLinearMap.NonlinearRightInverse.toFun f'symm (y - f x)
hg : g = fun x => x + ContinuousLinearMap.NonlinearRightInverse.toFun f'symm (y - f x)
u : ℕ → E := fun n => g^[n] b
hu : u = fun n => g^[n] b
usucc : ∀ (n : ℕ), u (n + 1) = g (u n)
⊢ y ∈ f '' closedBall b ε

Next tactic:
have A : ∀ z, dist (g z) z ≤ f'symm.nnnorm * dist (f z) y := by
    intro z
    rw [dist_eq_norm, hg, add_sub_cancel', dist_eq_norm']
    exact f'symm.bound _