theorem stereo_left_inv (hv : ‖v‖ = 1) {x : sphere (0 : E) 1} (hx : (x : E) ≠ v) :
    stereoInvFun hv (stereoToFun v x) = x := by
  ext
  simp only [stereoToFun_apply, stereoInvFun_apply, smul_add]
  -- name two frequently-occuring quantities and write down their basic properties
  set a : ℝ := innerSL _ v x
  set y := orthogonalProjection (ℝ ∙ v)ᗮ x
  have split : ↑x = a • v + ↑y := by
    convert (orthogonalProjection_add_orthogonalProjection_orthogonal (ℝ ∙ v) x).symm
    exact (orthogonalProjection_unit_singleton ℝ hv x).symm
  have hvy : ⟪v, y⟫_ℝ = 0 := Submodule.mem_orthogonal_singleton_iff_inner_right.mp y.2
  have pythag : 1 = a ^ 2 + ‖y‖ ^ 2 := by
    have hvy' : ⟪a • v, y⟫_ℝ = 0 := by simp only [inner_smul_left, hvy, mul_zero]
    convert norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero _ _ hvy' using 2
    -- Porting note: was simp [← split] but wasn't finding `norm_eq_of_mem_sphere`
    · simp only [norm_eq_of_mem_sphere, Nat.cast_one, mul_one, ← split]
    · simp [norm_smul, hv, ← sq, sq_abs]
    · exact sq _
  -- Porting note : added to work around cancel_denoms and nlinarith failures
  have duh : ‖y.val‖ ^ 2 = 1 - a ^ 2 := by
    rw [← Submodule.coe_norm, pythag]; ring
  -- two facts which will be helpful for clearing denominators in the main calculation
  have ha : 1 - a ≠ 0 := by
    have : a < 1 := (inner_lt_one_iff_real_of_norm_one hv (by simp)).mpr hx.symm
    linarith
  -- the core of the problem is these two algebraic identities:
  have h₁ : (2 ^ 2 / (1 - a) ^ 2 * ‖y‖ ^ 2 + 4)⁻¹ * 4 * (2 / (1 - a)) = 1 := by
    -- Porting note: used to be `field_simp; simp only [Submodule.coe_norm] at *; nlinarith`
    -- but cancel_denoms does not seem to be working and
    -- nlinarith cannot close the goal even if it did
    -- clear_value because field_simp does zeta-reduction (by design?) and is annoying
    clear_value a y
    field_simp
    rw [duh]
    ring
  have h₂ : (2 ^ 2 / (1 - a) ^ 2 * ‖y‖ ^ 2 + 4)⁻¹ * (2 ^ 2 / (1 - a) ^ 2 * ‖y‖ ^ 2 - 4) = a := by
    -- Porting note: field_simp is not behaving as in ml3
    -- see porting note above; previous proof used trans and was comparably complicated
    clear_value a y
    field_simp
    rw [duh]
    ring
  convert
    congr_arg₂ Add.add (congr_arg (fun t => t • (y : E)) h₁) (congr_arg (fun t => t • v) h₂) using 1
  · 

Proof state:
case h.e'_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
v : E
hv : ‖v‖ = 1
x : ↑(sphere 0 1)
hx : ↑x ≠ v
a : ℝ := ((innerSL ℝ) v) ↑x
y : ↥(Submodule.span ℝ {v})ᗮ := (orthogonalProjection (Submodule.span ℝ {v})ᗮ) ↑x
split : ↑x = a • v + ↑y
hvy : inner v ↑y = 0
pythag : 1 = a ^ 2 + ‖y‖ ^ 2
duh : ‖↑y‖ ^ 2 = 1 - a ^ 2
ha : 1 - a ≠ 0
h₁ : (2 ^ 2 / (1 - a) ^ 2 * ‖y‖ ^ 2 + 4)⁻¹ * 4 * (2 / (1 - a)) = 1
h₂ : (2 ^ 2 / (1 - a) ^ 2 * ‖y‖ ^ 2 + 4)⁻¹ * (2 ^ 2 / (1 - a) ^ 2 * ‖y‖ ^ 2 - 4) = a
⊢ (‖(2 / (1 - a)) • y‖ ^ 2 + 4)⁻¹ • 4 • ↑((2 / (1 - a)) • y) +
      (‖(2 / (1 - a)) • y‖ ^ 2 + 4)⁻¹ • (‖(2 / (1 - a)) • y‖ ^ 2 - 4) • v =
    Add.add ((fun t => t • ↑y) ((2 ^ 2 / (1 - a) ^ 2 * ‖y‖ ^ 2 + 4)⁻¹ * 4 * (2 / (1 - a))))
      ((fun t => t • v) ((2 ^ 2 / (1 - a) ^ 2 * ‖y‖ ^ 2 + 4)⁻¹ * (2 ^ 2 / (1 - a) ^ 2 * ‖y‖ ^ 2 - 4)))

Next tactic:
simp [inner_add_right, inner_smul_right, hvy, real_inner_self_eq_norm_mul_norm, hv, mul_smul,
      mul_pow, Real.norm_eq_abs, sq_abs, norm_smul]