set_option linter.unusedVariables false in
-- The proof here follows the argument in Hartshorne's Algebraic Geometry, Proposition II.2.2.
theorem toBasicOpen_surjective (f : R) : Function.Surjective (toBasicOpen R f) := by
  intro s
  -- In this proof, `basicOpen f` will play two distinct roles: Firstly, it is an open set in the
  -- prime spectrum. Secondly, it is used as an indexing type for various families of objects
  -- (open sets, ring elements, ...). In order to make the distinction clear, we introduce a type
  -- alias `ι` that is used whenever we want think of it as an indexing type.
  let ι : Type u := PrimeSpectrum.basicOpen f
  -- First, we pick some cover of basic opens, on which we can represent `s` as a fraction
  choose a' h' iDh' hxDh' s_eq' using locally_const_basicOpen R (PrimeSpectrum.basicOpen f) s
  -- Since basic opens are compact, we can pass to a finite subcover
  obtain ⟨t, ht_cover'⟩ :=
    (PrimeSpectrum.isCompact_basicOpen f).elim_finite_subcover
      (fun i : ι => PrimeSpectrum.basicOpen (h' i)) (fun i => PrimeSpectrum.isOpen_basicOpen)
      -- Here, we need to show that our basic opens actually form a cover of `basicOpen f`
      fun x hx => by rw [Set.mem_iUnion]; exact ⟨⟨x, hx⟩, hxDh' ⟨x, hx⟩⟩
  simp only [← Opens.coe_iSup, SetLike.coe_subset_coe] at ht_cover'
  -- We use the normalization lemma from above to obtain the relation `a i * h j = h i * a j`
  obtain ⟨a, h, iDh, ht_cover, ah_ha, s_eq⟩ :=
    normalize_finite_fraction_representation R (PrimeSpectrum.basicOpen f)
      s t a' h' iDh' ht_cover' s_eq'
  clear s_eq' iDh' hxDh' ht_cover' a' h'
  -- Porting note : simp with `[← SetLike.coe_subset_coe, Opens.coe_iSup]` does not result in
  -- desired form
  rw [← SetLike.coe_subset_coe, Opens.coe_iSup] at ht_cover
  replace ht_cover : (PrimeSpectrum.basicOpen f : Set <| PrimeSpectrum R) ⊆
    ⋃ (i : ι) (x : i ∈ t), (PrimeSpectrum.basicOpen (h i) : Set _)
  · convert ht_cover using 2
    exact funext fun j => by rw [Opens.coe_iSup]
  -- Next we show that some power of `f` is a linear combination of the `h i`
  obtain ⟨n, hn⟩ : f ∈ (Ideal.span (h '' ↑t)).radical := by
    rw [← PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radical, PrimeSpectrum.zeroLocus_span]
    -- Porting note : simp with `PrimeSpectrum.basicOpen_eq_zeroLocus_compl` does not work
    replace ht_cover : (PrimeSpectrum.zeroLocus {f})ᶜ ⊆
      ⋃ (i : ι) (x : i ∈ t), (PrimeSpectrum.zeroLocus {h i})ᶜ
    · convert ht_cover
      · rw [PrimeSpectrum.basicOpen_eq_zeroLocus_compl]
      · simp only [Opens.iSup_mk, Opens.carrier_eq_coe, PrimeSpectrum.basicOpen_eq_zeroLocus_compl]
    rw [Set.compl_subset_comm] at ht_cover
    -- Why doesn't `simp_rw` do this?
    simp_rw [Set.compl_iUnion, compl_compl, ← PrimeSpectrum.zeroLocus_iUnion,
      ← Finset.set_biUnion_coe, ← Set.image_eq_iUnion] at ht_cover
    apply PrimeSpectrum.vanishingIdeal_anti_mono ht_cover
    exact PrimeSpectrum.subset_vanishingIdeal_zeroLocus {f} (Set.mem_singleton f)
  replace hn := Ideal.mul_mem_left _ f hn
  erw [← pow_succ, Finsupp.mem_span_image_iff_total] at hn
  rcases hn with ⟨b, b_supp, hb⟩
  rw [Finsupp.total_apply_of_mem_supported R b_supp] at hb
  dsimp at hb
  -- Finally, we have all the ingredients.
  -- We claim that our preimage is given by `(∑ (i : ι) in t, b i * a i) / f ^ (n+1)`
  use
    IsLocalization.mk' (Localization.Away f) (∑ i : ι in t, b i * a i)
      (⟨f ^ (n + 1), n + 1, rfl⟩ : Submonoid.powers _)
  rw [toBasicOpen_mk']
  -- Since the structure sheaf is a sheaf, we can show the desired equality locally.
  -- Annoyingly, `Sheaf.eq_of_locally_eq'` requires an open cover indexed by a *type*, so we need to
  -- coerce our finset `t` to a type first.
  let tt := ((t : Set (PrimeSpectrum.basicOpen f)) : Type u)
  apply
    (structureSheaf R).eq_of_locally_eq' (fun i : tt => PrimeSpectrum.basicOpen (h i))
      (PrimeSpectrum.basicOpen f) fun i : tt => iDh i
  · -- This feels a little redundant, since already have `ht_cover` as a hypothesis
    -- Unfortunately, `ht_cover` uses a bounded union over the set `t`, while here we have the
    -- Union indexed by the type `tt`, so we need some boilerplate to translate one to the other
    intro x hx
    erw [TopologicalSpace.Opens.mem_iSup]
    have := ht_cover hx
    rw [← Finset.set_biUnion_coe, Set.mem_iUnion₂] at this
    rcases this with ⟨i, i_mem, x_mem⟩
    refine ⟨⟨i, i_mem⟩, x_mem⟩
  rintro ⟨i, hi⟩
  dsimp
  change (structureSheaf R).1.map _ _ = (structureSheaf R).1.map _ _
  rw [s_eq i hi, res_const]
  -- Again, `res_const` spits out an additional goal
  swap
  · intro y hy
    change y ∈ PrimeSpectrum.basicOpen (f ^ (n + 1))
    rw [PrimeSpectrum.basicOpen_pow f (n + 1) (by linarith)]
    exact (leOfHom (iDh i) : _) hy
  -- The rest of the proof is just computation
  apply const_ext
  rw [← hb, Finset.sum_mul, Finset.mul_sum]
  apply Finset.sum_congr rfl
  intro j hj
  rw [mul_assoc, ah_ha j hj i hi]
  

Proof state:
case h.h.mk.h
R : Type u
inst✝ : CommRing R
f : R
s : ↑((structureSheaf R).val.obj (op (PrimeSpectrum.basicOpen f)))
ι : Type u := ↥(PrimeSpectrum.basicOpen f)
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f
ah_ha : ∀ i ∈ t, ∀ j ∈ t, a i * h j = h i * a j
s_eq :
  ∀ i ∈ t,
    ((structureSheaf R).val.map (iDh i).op) s =
      const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))
        (_ : ∀ y ∈ PrimeSpectrum.basicOpen (h i), y ∈ PrimeSpectrum.basicOpen (h i))
ht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ i ∈ t, ↑(PrimeSpectrum.basicOpen (h i))
n : ℕ
b : ι →₀ R
b_supp : b ∈ Finsupp.supported R R ↑t
hb : ∑ i in t, b i * h i = f ^ (n + 1)
tt : Type u := ↑↑t
i : ι
hi : i ∈ ↑t
j : ↥(PrimeSpectrum.basicOpen f)
hj : j ∈ t
⊢ b j * (h j * a i) = a i * (b j * h j)

Next tactic:
ring