l3lab/ntp-mathlib huggingface dataset:
theorem normalize_finite_fraction_representation (U : Opens (PrimeSpectrum.Top R))
    (s : (structureSheaf R).1.obj (op U)) {ι : Type*} (t : Finset ι) (a h : ι → R)
    (iDh : ∀ i : ι, PrimeSpectrum.basicOpen (h i) ⟶ U)
    (h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i))
    (hs :
      ∀ i : ι,
        (const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) fun y hy => hy) =
          (structureSheaf R).1.map (iDh i).op s) :
    ∃ (a' h' : ι → R) (iDh' : ∀ i : ι, PrimeSpectrum.basicOpen (h' i) ⟶ U),
      (U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i)) ∧
        (∀ (i) (_ : i ∈ t) (j) (_ : j ∈ t), a' i * h' j = h' i * a' j) ∧
          ∀ i ∈ t,
            (structureSheaf R).1.map (iDh' i).op s =
              const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) fun y hy => hy := by
  -- First we show that the fractions `(a i * h j) / (h i * h j)` and `(h i * a j) / (h i * h j)`
  -- coincide in the localization of `R` at `h i * h j`
  have fractions_eq :
    ∀ i j : ι,
      IsLocalization.mk' (Localization.Away (h i * h j))
        (a i * h j) ⟨h i * h j, Submonoid.mem_powers _⟩ =
      IsLocalization.mk' _ (h i * a j) ⟨h i * h j, Submonoid.mem_powers _⟩ := by
    intro i j
    let D := PrimeSpectrum.basicOpen (h i * h j)
    

Proof state:
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs :
  ∀ (i : ι),
    const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))
        (_ : ∀ y ∈ PrimeSpectrum.basicOpen (h i), y ∈ PrimeSpectrum.basicOpen (h i)) =
      ((structureSheaf R).val.map (iDh i).op) s
i j : ι
D : Opens (PrimeSpectrum R) := PrimeSpectrum.basicOpen (h i * h j)
⊢ IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j)
      { val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) } =
    IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j)
      { val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) }

Next tactic:
let iDi : D ⟶ PrimeSpectrum.basicOpen (h i) := homOfLE (PrimeSpectrum.basicOpen_mul_le_left _ _)
theorem normalize_finite_fraction_representation (U : Opens (PrimeSpectrum.Top R))
    (s : (structureSheaf R).1.obj (op U)) {ι : Type*} (t : Finset ι) (a h : ι → R)
    (iDh : ∀ i : ι, PrimeSpectrum.basicOpen (h i) ⟶ U)
    (h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i))
    (hs :
      ∀ i : ι,
        (const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) fun y hy => hy) =
          (structureSheaf R).1.map (iDh i).op s) :
    ∃ (a' h' : ι → R) (iDh' : ∀ i : ι, PrimeSpectrum.basicOpen (h' i) ⟶ U),
      (U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i)) ∧
        (∀ (i) (_ : i ∈ t) (j) (_ : j ∈ t), a' i * h' j = h' i * a' j) ∧
          ∀ i ∈ t,
            (structureSheaf R).1.map (iDh' i).op s =
              const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) fun y hy => hy := by
  -- First we show that the fractions `(a i * h j) / (h i * h j)` and `(h i * a j) / (h i * h j)`
  -- coincide in the localization of `R` at `h i * h j`
  have fractions_eq :
    ∀ i j : ι,
      IsLocalization.mk' (Localization.Away (h i * h j))
        (a i * h j) ⟨h i * h j, Submonoid.mem_powers _⟩ =
      IsLocalization.mk' _ (h i * a j) ⟨h i * h j, Submonoid.mem_powers _⟩ := by
    intro i j
    let D := PrimeSpectrum.basicOpen (h i * h j)
    let iDi : D ⟶ PrimeSpectrum.basicOpen (h i) := homOfLE (PrimeSpectrum.basicOpen_mul_le_left _ _)
    

Proof state:
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs :
  ∀ (i : ι),
    const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))
        (_ : ∀ y ∈ PrimeSpectrum.basicOpen (h i), y ∈ PrimeSpectrum.basicOpen (h i)) =
      ((structureSheaf R).val.map (iDh i).op) s
i j : ι
D : Opens (PrimeSpectrum R) := PrimeSpectrum.basicOpen (h i * h j)
iDi : D ⟶ PrimeSpectrum.basicOpen (h i) :=
  homOfLE (_ : PrimeSpectrum.basicOpen (h i * h j) ≤ PrimeSpectrum.basicOpen (h i))
⊢ IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j)
      { val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) } =
    IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j)
      { val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) }

Next tactic:
let iDj : D ⟶ PrimeSpectrum.basicOpen (h j) :=
      homOfLE (PrimeSpectrum.basicOpen_mul_le_right _ _)
theorem normalize_finite_fraction_representation (U : Opens (PrimeSpectrum.Top R))
    (s : (structureSheaf R).1.obj (op U)) {ι : Type*} (t : Finset ι) (a h : ι → R)
    (iDh : ∀ i : ι, PrimeSpectrum.basicOpen (h i) ⟶ U)
    (h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i))
    (hs :
      ∀ i : ι,
        (const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) fun y hy => hy) =
          (structureSheaf R).1.map (iDh i).op s) :
    ∃ (a' h' : ι → R) (iDh' : ∀ i : ι, PrimeSpectrum.basicOpen (h' i) ⟶ U),
      (U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i)) ∧
        (∀ (i) (_ : i ∈ t) (j) (_ : j ∈ t), a' i * h' j = h' i * a' j) ∧
          ∀ i ∈ t,
            (structureSheaf R).1.map (iDh' i).op s =
              const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) fun y hy => hy := by
  -- First we show that the fractions `(a i * h j) / (h i * h j)` and `(h i * a j) / (h i * h j)`
  -- coincide in the localization of `R` at `h i * h j`
  have fractions_eq :
    ∀ i j : ι,
      IsLocalization.mk' (Localization.Away (h i * h j))
        (a i * h j) ⟨h i * h j, Submonoid.mem_powers _⟩ =
      IsLocalization.mk' _ (h i * a j) ⟨h i * h j, Submonoid.mem_powers _⟩ := by
    intro i j
    let D := PrimeSpectrum.basicOpen (h i * h j)
    let iDi : D ⟶ PrimeSpectrum.basicOpen (h i) := homOfLE (PrimeSpectrum.basicOpen_mul_le_left _ _)
    let iDj : D ⟶ PrimeSpectrum.basicOpen (h j) :=
      homOfLE (PrimeSpectrum.basicOpen_mul_le_right _ _)
    -- Crucially, we need injectivity of `toBasicOpen`
    

Proof state:
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs :
  ∀ (i : ι),
    const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))
        (_ : ∀ y ∈ PrimeSpectrum.basicOpen (h i), y ∈ PrimeSpectrum.basicOpen (h i)) =
      ((structureSheaf R).val.map (iDh i).op) s
i j : ι
D : Opens (PrimeSpectrum R) := PrimeSpectrum.basicOpen (h i * h j)
iDi : D ⟶ PrimeSpectrum.basicOpen (h i) :=
  homOfLE (_ : PrimeSpectrum.basicOpen (h i * h j) ≤ PrimeSpectrum.basicOpen (h i))
iDj : D ⟶ PrimeSpectrum.basicOpen (h j) :=
  homOfLE (_ : PrimeSpectrum.basicOpen (h i * h j) ≤ PrimeSpectrum.basicOpen (h j))
⊢ IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j)
      { val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) } =
    IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j)
      { val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) }

Next tactic:
apply toBasicOpen_injective R (h i * h j)
theorem normalize_finite_fraction_representation (U : Opens (PrimeSpectrum.Top R))
    (s : (structureSheaf R).1.obj (op U)) {ι : Type*} (t : Finset ι) (a h : ι → R)
    (iDh : ∀ i : ι, PrimeSpectrum.basicOpen (h i) ⟶ U)
    (h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i))
    (hs :
      ∀ i : ι,
        (const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) fun y hy => hy) =
          (structureSheaf R).1.map (iDh i).op s) :
    ∃ (a' h' : ι → R) (iDh' : ∀ i : ι, PrimeSpectrum.basicOpen (h' i) ⟶ U),
      (U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i)) ∧
        (∀ (i) (_ : i ∈ t) (j) (_ : j ∈ t), a' i * h' j = h' i * a' j) ∧
          ∀ i ∈ t,
            (structureSheaf R).1.map (iDh' i).op s =
              const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) fun y hy => hy := by
  -- First we show that the fractions `(a i * h j) / (h i * h j)` and `(h i * a j) / (h i * h j)`
  -- coincide in the localization of `R` at `h i * h j`
  have fractions_eq :
    ∀ i j : ι,
      IsLocalization.mk' (Localization.Away (h i * h j))
        (a i * h j) ⟨h i * h j, Submonoid.mem_powers _⟩ =
      IsLocalization.mk' _ (h i * a j) ⟨h i * h j, Submonoid.mem_powers _⟩ := by
    intro i j
    let D := PrimeSpectrum.basicOpen (h i * h j)
    let iDi : D ⟶ PrimeSpectrum.basicOpen (h i) := homOfLE (PrimeSpectrum.basicOpen_mul_le_left _ _)
    let iDj : D ⟶ PrimeSpectrum.basicOpen (h j) :=
      homOfLE (PrimeSpectrum.basicOpen_mul_le_right _ _)
    -- Crucially, we need injectivity of `toBasicOpen`
    apply toBasicOpen_injective R (h i * h j)
    

Proof state:
case a
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs :
  ∀ (i : ι),
    const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))
        (_ : ∀ y ∈ PrimeSpectrum.basicOpen (h i), y ∈ PrimeSpectrum.basicOpen (h i)) =
      ((structureSheaf R).val.map (iDh i).op) s
i j : ι
D : Opens (PrimeSpectrum R) := PrimeSpectrum.basicOpen (h i * h j)
iDi : D ⟶ PrimeSpectrum.basicOpen (h i) :=
  homOfLE (_ : PrimeSpectrum.basicOpen (h i * h j) ≤ PrimeSpectrum.basicOpen (h i))
iDj : D ⟶ PrimeSpectrum.basicOpen (h j) :=
  homOfLE (_ : PrimeSpectrum.basicOpen (h i * h j) ≤ PrimeSpectrum.basicOpen (h j))
⊢ (toBasicOpen R (h i * h j))
      (IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j)
        { val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) }) =
    (toBasicOpen R (h i * h j))
      (IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j)
        { val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) })

Next tactic:
rw [toBasicOpen_mk', toBasicOpen_mk']
theorem normalize_finite_fraction_representation (U : Opens (PrimeSpectrum.Top R))
    (s : (structureSheaf R).1.obj (op U)) {ι : Type*} (t : Finset ι) (a h : ι → R)
    (iDh : ∀ i : ι, PrimeSpectrum.basicOpen (h i) ⟶ U)
    (h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i))
    (hs :
      ∀ i : ι,
        (const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) fun y hy => hy) =
          (structureSheaf R).1.map (iDh i).op s) :
    ∃ (a' h' : ι → R) (iDh' : ∀ i : ι, PrimeSpectrum.basicOpen (h' i) ⟶ U),
      (U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i)) ∧
        (∀ (i) (_ : i ∈ t) (j) (_ : j ∈ t), a' i * h' j = h' i * a' j) ∧
          ∀ i ∈ t,
            (structureSheaf R).1.map (iDh' i).op s =
              const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) fun y hy => hy := by
  -- First we show that the fractions `(a i * h j) / (h i * h j)` and `(h i * a j) / (h i * h j)`
  -- coincide in the localization of `R` at `h i * h j`
  have fractions_eq :
    ∀ i j : ι,
      IsLocalization.mk' (Localization.Away (h i * h j))
        (a i * h j) ⟨h i * h j, Submonoid.mem_powers _⟩ =
      IsLocalization.mk' _ (h i * a j) ⟨h i * h j, Submonoid.mem_powers _⟩ := by
    intro i j
    let D := PrimeSpectrum.basicOpen (h i * h j)
    let iDi : D ⟶ PrimeSpectrum.basicOpen (h i) := homOfLE (PrimeSpectrum.basicOpen_mul_le_left _ _)
    let iDj : D ⟶ PrimeSpectrum.basicOpen (h j) :=
      homOfLE (PrimeSpectrum.basicOpen_mul_le_right _ _)
    -- Crucially, we need injectivity of `toBasicOpen`
    apply toBasicOpen_injective R (h i * h j)
    rw [toBasicOpen_mk', toBasicOpen_mk']
    

Proof state:
case a
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs :
  ∀ (i : ι),
    const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))
        (_ : ∀ y ∈ PrimeSpectrum.basicOpen (h i), y ∈ PrimeSpectrum.basicOpen (h i)) =
      ((structureSheaf R).val.map (iDh i).op) s
i j : ι
D : Opens (PrimeSpectrum R) := PrimeSpectrum.basicOpen (h i * h j)
iDi : D ⟶ PrimeSpectrum.basicOpen (h i) :=
  homOfLE (_ : PrimeSpectrum.basicOpen (h i * h j) ≤ PrimeSpectrum.basicOpen (h i))
iDj : D ⟶ PrimeSpectrum.basicOpen (h j) :=
  homOfLE (_ : PrimeSpectrum.basicOpen (h i * h j) ≤ PrimeSpectrum.basicOpen (h j))
⊢ const R (a i * h j) (↑{ val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) })
      (PrimeSpectrum.basicOpen (h i * h j))
      (_ :
        ∀ x ∈ PrimeSpectrum.basicOpen (h i * h j),
          ↑{ val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) } ∈
            Ideal.primeCompl x.asIdeal) =
    const R (h i * a j) (↑{ val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) })
      (PrimeSpectrum.basicOpen (h i * h j))
      (_ :
        ∀ x ∈ PrimeSpectrum.basicOpen (h i * h j),
          ↑{ val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) } ∈
            Ideal.primeCompl x.asIdeal)

Next tactic:
simp only []
theorem normalize_finite_fraction_representation (U : Opens (PrimeSpectrum.Top R))
    (s : (structureSheaf R).1.obj (op U)) {ι : Type*} (t : Finset ι) (a h : ι → R)
    (iDh : ∀ i : ι, PrimeSpectrum.basicOpen (h i) ⟶ U)
    (h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i))
    (hs :
      ∀ i : ι,
        (const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) fun y hy => hy) =
          (structureSheaf R).1.map (iDh i).op s) :
    ∃ (a' h' : ι → R) (iDh' : ∀ i : ι, PrimeSpectrum.basicOpen (h' i) ⟶ U),
      (U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i)) ∧
        (∀ (i) (_ : i ∈ t) (j) (_ : j ∈ t), a' i * h' j = h' i * a' j) ∧
          ∀ i ∈ t,
            (structureSheaf R).1.map (iDh' i).op s =
              const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) fun y hy => hy := by
  -- First we show that the fractions `(a i * h j) / (h i * h j)` and `(h i * a j) / (h i * h j)`
  -- coincide in the localization of `R` at `h i * h j`
  have fractions_eq :
    ∀ i j : ι,
      IsLocalization.mk' (Localization.Away (h i * h j))
        (a i * h j) ⟨h i * h j, Submonoid.mem_powers _⟩ =
      IsLocalization.mk' _ (h i * a j) ⟨h i * h j, Submonoid.mem_powers _⟩ := by
    intro i j
    let D := PrimeSpectrum.basicOpen (h i * h j)
    let iDi : D ⟶ PrimeSpectrum.basicOpen (h i) := homOfLE (PrimeSpectrum.basicOpen_mul_le_left _ _)
    let iDj : D ⟶ PrimeSpectrum.basicOpen (h j) :=
      homOfLE (PrimeSpectrum.basicOpen_mul_le_right _ _)
    -- Crucially, we need injectivity of `toBasicOpen`
    apply toBasicOpen_injective R (h i * h j)
    rw [toBasicOpen_mk', toBasicOpen_mk']
    simp only []
    -- Here, both sides of the equation are equal to a restriction of `s`
    

Proof state:
case a
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs :
  ∀ (i : ι),
    const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))
        (_ : ∀ y ∈ PrimeSpectrum.basicOpen (h i), y ∈ PrimeSpectrum.basicOpen (h i)) =
      ((structureSheaf R).val.map (iDh i).op) s
i j : ι
D : Opens (PrimeSpectrum R) := PrimeSpectrum.basicOpen (h i * h j)
iDi : D ⟶ PrimeSpectrum.basicOpen (h i) :=
  homOfLE (_ : PrimeSpectrum.basicOpen (h i * h j) ≤ PrimeSpectrum.basicOpen (h i))
iDj : D ⟶ PrimeSpectrum.basicOpen (h j) :=
  homOfLE (_ : PrimeSpectrum.basicOpen (h i * h j) ≤ PrimeSpectrum.basicOpen (h j))
⊢ const R (a i * h j) (h i * h j) (PrimeSpectrum.basicOpen (h i * h j))
      (_ :
        ∀ x ∈ PrimeSpectrum.basicOpen (h i * h j),
          ↑{ val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) } ∈
            Ideal.primeCompl x.asIdeal) =
    const R (h i * a j) (h i * h j) (PrimeSpectrum.basicOpen (h i * h j))
      (_ :
        ∀ x ∈ PrimeSpectrum.basicOpen (h i * h j),
          ↑{ val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) } ∈
            Ideal.primeCompl x.asIdeal)

Next tactic:
trans
theorem normalize_finite_fraction_representation (U : Opens (PrimeSpectrum.Top R))
    (s : (structureSheaf R).1.obj (op U)) {ι : Type*} (t : Finset ι) (a h : ι → R)
    (iDh : ∀ i : ι, PrimeSpectrum.basicOpen (h i) ⟶ U)
    (h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i))
    (hs :
      ∀ i : ι,
        (const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) fun y hy => hy) =
          (structureSheaf R).1.map (iDh i).op s) :
    ∃ (a' h' : ι → R) (iDh' : ∀ i : ι, PrimeSpectrum.basicOpen (h' i) ⟶ U),
      (U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i)) ∧
        (∀ (i) (_ : i ∈ t) (j) (_ : j ∈ t), a' i * h' j = h' i * a' j) ∧
          ∀ i ∈ t,
            (structureSheaf R).1.map (iDh' i).op s =
              const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) fun y hy => hy := by
  -- First we show that the fractions `(a i * h j) / (h i * h j)` and `(h i * a j) / (h i * h j)`
  -- coincide in the localization of `R` at `h i * h j`
  have fractions_eq :
    ∀ i j : ι,
      IsLocalization.mk' (Localization.Away (h i * h j))
        (a i * h j) ⟨h i * h j, Submonoid.mem_powers _⟩ =
      IsLocalization.mk' _ (h i * a j) ⟨h i * h j, Submonoid.mem_powers _⟩ := by
    intro i j
    let D := PrimeSpectrum.basicOpen (h i * h j)
    let iDi : D ⟶ PrimeSpectrum.basicOpen (h i) := homOfLE (PrimeSpectrum.basicOpen_mul_le_left _ _)
    let iDj : D ⟶ PrimeSpectrum.basicOpen (h j) :=
      homOfLE (PrimeSpectrum.basicOpen_mul_le_right _ _)
    -- Crucially, we need injectivity of `toBasicOpen`
    apply toBasicOpen_injective R (h i * h j)
    rw [toBasicOpen_mk', toBasicOpen_mk']
    simp only []
    -- Here, both sides of the equation are equal to a restriction of `s`
    trans
    convert congr_arg ((structureSheaf R).1.map iDj.op) (hs j).symm using 1
    convert congr_arg ((structureSheaf R).1.map iDi.op) (hs i) using 1; swap
    all_goals 

Proof state:
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs :
  ∀ (i : ι),
    const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))
        (_ : ∀ y ∈ PrimeSpectrum.basicOpen (h i), y ∈ PrimeSpectrum.basicOpen (h i)) =
      ((structureSheaf R).val.map (iDh i).op) s
i j : ι
D : Opens (PrimeSpectrum R) := PrimeSpectrum.basicOpen (h i * h j)
iDi : D ⟶ PrimeSpectrum.basicOpen (h i) :=
  homOfLE (_ : PrimeSpectrum.basicOpen (h i * h j) ≤ PrimeSpectrum.basicOpen (h i))
iDj : D ⟶ PrimeSpectrum.basicOpen (h j) :=
  homOfLE (_ : PrimeSpectrum.basicOpen (h i * h j) ≤ PrimeSpectrum.basicOpen (h j))
⊢ ((structureSheaf R).val.map iDj.op)
      (const R (a j) (h j) (PrimeSpectrum.basicOpen (h j))
        (_ : ∀ y ∈ PrimeSpectrum.basicOpen (h j), y ∈ PrimeSpectrum.basicOpen (h j))) =
    const R (h i * a j) (h i * h j) (PrimeSpectrum.basicOpen (h i * h j))
      (_ :
        ∀ x ∈ PrimeSpectrum.basicOpen (h i * h j),
          ↑{ val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) } ∈
            Ideal.primeCompl x.asIdeal)

Next tactic:
rw [res_const]
theorem normalize_finite_fraction_representation (U : Opens (PrimeSpectrum.Top R))
    (s : (structureSheaf R).1.obj (op U)) {ι : Type*} (t : Finset ι) (a h : ι → R)
    (iDh : ∀ i : ι, PrimeSpectrum.basicOpen (h i) ⟶ U)
    (h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i))
    (hs :
      ∀ i : ι,
        (const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) fun y hy => hy) =
          (structureSheaf R).1.map (iDh i).op s) :
    ∃ (a' h' : ι → R) (iDh' : ∀ i : ι, PrimeSpectrum.basicOpen (h' i) ⟶ U),
      (U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i)) ∧
        (∀ (i) (_ : i ∈ t) (j) (_ : j ∈ t), a' i * h' j = h' i * a' j) ∧
          ∀ i ∈ t,
            (structureSheaf R).1.map (iDh' i).op s =
              const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) fun y hy => hy := by
  -- First we show that the fractions `(a i * h j) / (h i * h j)` and `(h i * a j) / (h i * h j)`
  -- coincide in the localization of `R` at `h i * h j`
  have fractions_eq :
    ∀ i j : ι,
      IsLocalization.mk' (Localization.Away (h i * h j))
        (a i * h j) ⟨h i * h j, Submonoid.mem_powers _⟩ =
      IsLocalization.mk' _ (h i * a j) ⟨h i * h j, Submonoid.mem_powers _⟩ := by
    intro i j
    let D := PrimeSpectrum.basicOpen (h i * h j)
    let iDi : D ⟶ PrimeSpectrum.basicOpen (h i) := homOfLE (PrimeSpectrum.basicOpen_mul_le_left _ _)
    let iDj : D ⟶ PrimeSpectrum.basicOpen (h j) :=
      homOfLE (PrimeSpectrum.basicOpen_mul_le_right _ _)
    -- Crucially, we need injectivity of `toBasicOpen`
    apply toBasicOpen_injective R (h i * h j)
    rw [toBasicOpen_mk', toBasicOpen_mk']
    simp only []
    -- Here, both sides of the equation are equal to a restriction of `s`
    trans
    convert congr_arg ((structureSheaf R).1.map iDj.op) (hs j).symm using 1
    convert congr_arg ((structureSheaf R).1.map iDi.op) (hs i) using 1; swap
    all_goals 

Proof state:
case h.e'_2
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs :
  ∀ (i : ι),
    const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))
        (_ : ∀ y ∈ PrimeSpectrum.basicOpen (h i), y ∈ PrimeSpectrum.basicOpen (h i)) =
      ((structureSheaf R).val.map (iDh i).op) s
i j : ι
D : Opens (PrimeSpectrum R) := PrimeSpectrum.basicOpen (h i * h j)
iDi : D ⟶ PrimeSpectrum.basicOpen (h i) :=
  homOfLE (_ : PrimeSpectrum.basicOpen (h i * h j) ≤ PrimeSpectrum.basicOpen (h i))
iDj : D ⟶ PrimeSpectrum.basicOpen (h j) :=
  homOfLE (_ : PrimeSpectrum.basicOpen (h i * h j) ≤ PrimeSpectrum.basicOpen (h j))
⊢ const R (a i * h j) (h i * h j) (PrimeSpectrum.basicOpen (h i * h j))
      (_ :
        ∀ x ∈ PrimeSpectrum.basicOpen (h i * h j),
          ↑{ val := h i * h j, property := (_ : h i * h j ∈ Submonoid.powers (h i * h j)) } ∈
            Ideal.primeCompl x.asIdeal) =
    ((structureSheaf R).val.map iDi.op)
      (const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))
        (_ : ∀ y ∈ PrimeSpectrum.basicOpen (h i), y ∈ PrimeSpectrum.basicOpen (h i)))

Next tactic:
rw [res_const]
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
  

Proof state:
R : Type u
inst✝ : CommRing R
f : R
⊢ Function.Surjective ⇑(toBasicOpen R f)

Next tactic:
intro s
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.
  

Proof state:
R : Type u
inst✝ : CommRing R
f : R
s : ↑((structureSheaf R).val.obj (op (PrimeSpectrum.basicOpen f)))
⊢ ∃ a, (toBasicOpen R f) a = s

Next tactic:
let ι : Type u := PrimeSpectrum.basicOpen f