awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (goal)
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
s : R
f : M
n : ℕ
⊢ HSMul.hSMul (β :=
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen s)))
      (s ^ n)
      ((IsLocalizedModule.lift (Submonoid.powers s) (LocalizedModule.mkLinearMap (Submonoid.powers s) M)
              (toOpenₗ R M (basicOpen s)) ⋯ :
            (a : LocalizedModule.Away s M) → (structureSheafInType R M).obj.obj (op (basicOpen s)))
          (LocalizedModule.mk f
            (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) (Submonoid.powers s) x) (s ^ n) ⋯)) :
        Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
          (ObjectProperty.FullSubcategory.obj (P :=
            CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
            (structureSheafInType R M))
          (op (basicOpen s))) =
    HSMul.hSMul (β :=
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen s)))
      (s ^ n) (const f (s ^ n) (basicOpen s) ⋯)
have rw : Eq (α :=
  Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op (basicOpen s)))
  ((toOpenₗ R M (basicOpen s) : (a : M) → (structureSheafInType R M).obj.obj (op (basicOpen s))) f :
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen s)))
  (const (s ^ n • f) (s ^ n) (basicOpen s) sorry :
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen s)))
Tactic: rw [← map_smul, LocalizedModule.smul'_mk, ← Submonoid.mk_smul (S := .powers s) _ ⟨n, rfl⟩,
    LocalizedModule.mk_cancel, ← LocalizedModule.mkLinearMap_apply, IsLocalizedModule.lift_apply,
    smul_const]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (goal)
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
f : R
a : M
b : ↥(Submonoid.powers f)
c : M
d : ↥(Submonoid.powers f)
h_eq :
  Eq (α :=
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen f)))
    ((toBasicOpenₗ R M f : (a : LocalizedModule.Away f M) → (structureSheafInType R M).obj.obj (op (basicOpen f)))
        (LocalizedModule.mk a b) :
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen f)))
    ((toBasicOpenₗ R M f : (a : LocalizedModule.Away f M) → (structureSheafInType R M).obj.obj (op (basicOpen f)))
        (LocalizedModule.mk c d) :
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen f)))
this : Membership.mem (γ := Ideal R) (Ideal.radical (Submodule.colon (M := M) ⊥ {d • a - b • c})) f
⊢ Eq (α := LocalizedModule.Away f M) (LocalizedModule.mk a b) (LocalizedModule.mk c d)
have rw : ∃ (u : ↥(Submonoid.powers f)), u • d • a = u • b • c
Tactic: rw [LocalizedModule.mk_eq]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (goal)
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
f : R
a : M
b : ↥(Submonoid.powers f)
c : M
d : ↥(Submonoid.powers f)
h_eq :
  Eq (α :=
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen f)))
    ((toBasicOpenₗ R M f : (a : LocalizedModule.Away f M) → (structureSheafInType R M).obj.obj (op (basicOpen f)))
        (LocalizedModule.mk a b) :
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen f)))
    ((toBasicOpenₗ R M f : (a : LocalizedModule.Away f M) → (structureSheafInType R M).obj.obj (op (basicOpen f)))
        (LocalizedModule.mk c d) :
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen f)))
this : Membership.mem (γ := Ideal R) (Ideal.radical (Submodule.colon (M := M) ⊥ {d • a - b • c})) f
⊢ ∃ (u : ↥(Submonoid.powers f)), u • d • a = u • b • c
have obtain : ∀ (n : ℕ) (hn : Membership.mem (γ := Ideal R) (Submodule.colon (M := M) ⊥ {d • a - b • c}) (f ^ n)),
  ∃ (u : ↥(Submonoid.powers f)), u • d • a = u • b • c
Tactic: obtain ⟨n, hn⟩ := this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (goal)
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
f : R
a : M
b : ↥(Submonoid.powers f)
c : M
d : ↥(Submonoid.powers f)
h_eq :
  Eq (α :=
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen f)))
    ((toBasicOpenₗ R M f : (a : LocalizedModule.Away f M) → (structureSheafInType R M).obj.obj (op (basicOpen f)))
        (LocalizedModule.mk a b) :
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen f)))
    ((toBasicOpenₗ R M f : (a : LocalizedModule.Away f M) → (structureSheafInType R M).obj.obj (op (basicOpen f)))
        (LocalizedModule.mk c d) :
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen f)))
⊢ Eq (α := LocalizedModule.Away f M) (LocalizedModule.mk a b) (LocalizedModule.mk c d)
have h : Membership.mem (γ := Ideal R) (Ideal.radical (Submodule.colon (M := M) ⊥ {d • a - b • c})) f
Tactic: suffices f ∈ ((⊥ : Submodule R M).colon {d • a - b • c}).radical by
    rw [LocalizedModule.mk_eq]
    obtain ⟨n, hn⟩ := this
    exact ⟨⟨f ^ n, n, rfl⟩, by simpa [sub_eq_zero, smul_sub] using Submodule.mem_colon.mp hn _ rfl⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (hypothesis)
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
f : R
a : M
b : ↥(Submonoid.powers f)
c : M
d : ↥(Submonoid.powers f)
h_eq :
  Eq (α :=
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen f)))
    ((toBasicOpenₗ R M f : (a : LocalizedModule.Away f M) → (structureSheafInType R M).obj.obj (op (basicOpen f)))
        (LocalizedModule.mk a b) :
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen f)))
    ((toBasicOpenₗ R M f : (a : LocalizedModule.Away f M) → (structureSheafInType R M).obj.obj (op (basicOpen f)))
        (LocalizedModule.mk c d) :
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen f)))
⊢ Membership.mem (γ := Ideal R) (Ideal.radical (Submodule.colon (M := M) ⊥ {d • a - b • c})) f
have h_eq₁ : Eq (α :=
  Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op (basicOpen f)))
  (const a (↑b : R) (basicOpen f) sorry) (const c (↑d : R) (basicOpen f) sorry)
Tactic: simp only [toBasicOpenₗ_mk] at h_eq
Cleared hypotheses: h_eq
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (goal)
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
f : R
a : M
b : ↥(Submonoid.powers f)
c : M
d : ↥(Submonoid.powers f)
h_eq :
  Eq (α :=
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen f)))
    (const a (↑b : R) (basicOpen f) ⋯) (const c (↑d : R) (basicOpen f) ⋯)
⊢ Membership.mem (γ := Ideal R) (Ideal.radical (Submodule.colon (M := M) ⊥ {d • a - b • c})) f
have h : ∀ (x : PrimeSpectrum R),
  Membership.mem (γ := Set (PrimeSpectrum R))
      (PrimeSpectrum.zeroLocus (↑(Submodule.colon (M := M) ⊥ {d • a - b • c}) : Set R)) x →
    f ∈ x.asIdeal
Tactic: rw [← PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radical, PrimeSpectrum.mem_vanishingIdeal]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (goal)
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
f : R
a : M
b : ↥(Submonoid.powers f)
c : M
d : ↥(Submonoid.powers f)
h_eq :
  Eq (α :=
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen f)))
    (const a (↑b : R) (basicOpen f) ⋯) (const c (↑d : R) (basicOpen f) ⋯)
p : PrimeSpectrum R
hfp :
  Membership.mem (γ := Set (PrimeSpectrum R))
    (PrimeSpectrum.zeroLocus (↑(Submodule.colon (M := M) ⊥ {d • a - b • c}) : Set R)) p
⊢ f ∈ p.asIdeal
have h : ∀ (hfp : f ∉ p.asIdeal), p ∉ PrimeSpectrum.zeroLocus (↑(Submodule.colon (M := M) ⊥ {d • a - b • c}) : Set R)
Tactic: contrapose hfp
Cleared hypotheses: hfp
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (goal)
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
f : R
a : M
b : ↥(Submonoid.powers f)
c : M
d : ↥(Submonoid.powers f)
h_eq :
  Eq (α :=
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen f)))
    (const a (↑b : R) (basicOpen f) ⋯) (const c (↑d : R) (basicOpen f) ⋯)
p : PrimeSpectrum R
hfp : f ∉ p.asIdeal
⊢ p ∉ PrimeSpectrum.zeroLocus (↑(Submodule.colon (M := M) ⊥ {d • a - b • c}) : Set R)
have h : ∀
  (u :
    ↥(↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
                Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen f))) x) p hfp) :
            (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl)
  (hu :
    u •
        HSMul.hSMul (α :=
          ↥(↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
                      Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen f))) x) p
                      hfp) :
                  (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl)
          (Subtype.mk (p := fun (x : R) ↦
            x ∈
              (↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
                        Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen f))) x) p
                        hfp) :
                    (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl)
            (↑d : R) sorry)
          a =
      u •
        HSMul.hSMul (α :=
          ↥(↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
                      Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen f))) x) p
                      hfp) :
                  (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl)
          (Subtype.mk (p := fun (x : R) ↦
            x ∈
              (↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
                        Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen f))) x) p
                        hfp) :
                    (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl)
            (↑b : R) sorry)
          c),
  p ∉ PrimeSpectrum.zeroLocus (↑(Submodule.colon (M := M) ⊥ {d • a - b • c}) : Set R)
Tactic: obtain ⟨u, hu⟩ := LocalizedModule.mk_eq.mp congr(($h_eq).1 ⟨p, hfp⟩)
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (goal)
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
f : R
a : M
b : ↥(Submonoid.powers f)
c : M
d : ↥(Submonoid.powers f)
h_eq :
  Eq (α :=
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen f)))
    (const a (↑b : R) (basicOpen f) ⋯) (const c (↑d : R) (basicOpen f) ⋯)
p : PrimeSpectrum R
hfp : f ∉ p.asIdeal
u :
  ↥(↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
              Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen f))) x) p hfp) :
          (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl
hu :
  u •
      HSMul.hSMul (α :=
        ↥(↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
                    Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen f))) x) p hfp) :
                (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl)
        (Subtype.mk (p := fun (x : R) ↦
          x ∈
            (↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
                      Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen f))) x) p
                      hfp) :
                  (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl)
          (↑d : R) ⋯)
        a =
    u •
      HSMul.hSMul (α :=
        ↥(↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
                    Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen f))) x) p hfp) :
                (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl)
        (Subtype.mk (p := fun (x : R) ↦
          x ∈
            (↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
                      Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen f))) x) p
                      hfp) :
                  (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl)
          (↑b : R) ⋯)
        c
⊢ p ∉ PrimeSpectrum.zeroLocus (↑(Submodule.colon (M := M) ⊥ {d • a - b • c}) : Set R)
have h : ∃ a_1 ∈ (↑(Submodule.colon (M := M) ⊥ {d • a - b • c}) : Set R), a_1 ∉ (↑p.asIdeal : Set R)
Tactic: rw [PrimeSpectrum.mem_zeroLocus, Set.not_subset]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (hypothesis)
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
hU : IsCompact (↑U : Set (↑(PrimeSpectrum.Top R) : Type u))
s :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
g : (x : ↥U) → R
hxg :
  ∀ (x : ↥U), Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen (g x)) (↑x : (↑(PrimeSpectrum.Top R) : Type u))
igU : ∀ (x : ↥U), basicOpen (g x) ≤ U
f : (x : ↥U) → M
H :
  ∀ (x : ↥U),
    Eq (α :=
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen (g x))))
      (const (f x) (g x) (basicOpen (g x)) ⋯)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              ⋯.hom.op) :
          (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g x)))))
        s)
i j : ↥U
⊢ Eq (α :=
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen (g i * g j))))
    (const (g i • f j) (g i * g j) (basicOpen (g i * g j)) ⋯) (const (g j • f i) (g i * g j) (basicOpen (g i * g j)) ⋯)
have this : Eq (α :=
  Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op (basicOpen (g i))))
  (const (f i) (g i) (basicOpen (g i)) sorry)
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
        (CategoryTheory.Functor.map
          (ObjectProperty.FullSubcategory.obj (P :=
            CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
            (structureSheafInType R M))
          sorry.hom.op) :
      (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
        (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g i)))))
    s)
Tactic: have := H i
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (goal)
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
hU : IsCompact (↑U : Set (↑(PrimeSpectrum.Top R) : Type u))
s :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
g : (x : ↥U) → R
hxg :
  ∀ (x : ↥U), Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen (g x)) (↑x : (↑(PrimeSpectrum.Top R) : Type u))
igU : ∀ (x : ↥U), basicOpen (g x) ≤ U
f : (x : ↥U) → M
H :
  ∀ (x : ↥U),
    Eq (α :=
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen (g x))))
      (const (f x) (g x) (basicOpen (g x)) ⋯)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              ⋯.hom.op) :
          (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g x)))))
        s)
n : (i j : ↥U) → ℕ
hn : ∀ (i j : ↥U), ((g i * g j) ^ (n i j + 1) * g i) • f j = ((g i * g j) ^ (n i j + 1) * g j) • f i
t : Finset ↥U
ht : (↑U : Set (↑(PrimeSpectrum.Top R) : Type u)) ⊆ ⋃ i ∈ t, (↑(basicOpen (g i)) : Set (PrimeSpectrum R))
N : ℕ := (t ×ˢ t).sup fun (x : ↥U × ↥U) ↦ n x.1 x.2 + 1
i j : ↥t
⊢ g (↑j : ↥U) ^ (N + 1) • g (↑i : ↥U) ^ N • f (↑i : ↥U) = g (↑i : ↥U) ^ (N + 1) • g (↑j : ↥U) ^ N • f (↑j : ↥U)
have h : g (↑j : ↥U) ^ (N + 1) • g (↑i : ↥U) ^ N • f (↑i : ↥U) = (g (↑i : ↥U) * g (↑j : ↥U)) ^ N • g (↑j : ↥U) • f (↑i : ↥U)
Tactic: convert_to (g i * g ↑j) ^ N • g j • f i = (g i * g ↑j) ^ N • g i • f j
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (goal)
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
hU : IsCompact (↑U : Set (↑(PrimeSpectrum.Top R) : Type u))
s :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
g : (x : ↥U) → R
hxg :
  ∀ (x : ↥U), Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen (g x)) (↑x : (↑(PrimeSpectrum.Top R) : Type u))
igU : ∀ (x : ↥U), basicOpen (g x) ≤ U
f : (x : ↥U) → M
H :
  ∀ (x : ↥U),
    Eq (α :=
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen (g x))))
      (const (f x) (g x) (basicOpen (g x)) ⋯)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              ⋯.hom.op) :
          (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g x)))))
        s)
n : (i j : ↥U) → ℕ
hn : ∀ (i j : ↥U), ((g i * g j) ^ (n i j + 1) * g i) • f j = ((g i * g j) ^ (n i j + 1) * g j) • f i
t : Finset ↥U
ht : (↑U : Set (↑(PrimeSpectrum.Top R) : Type u)) ⊆ ⋃ i ∈ t, (↑(basicOpen (g i)) : Set (PrimeSpectrum R))
N : ℕ := (t ×ˢ t).sup fun (x : ↥U × ↥U) ↦ n x.1 x.2 + 1
i j : ↥t
h : g (↑j : ↥U) ^ (N + 1) • g (↑i : ↥U) ^ N • f (↑i : ↥U) = (g (↑i : ↥U) * g (↑j : ↥U)) ^ N • g (↑j : ↥U) • f (↑i : ↥U)
⊢ g (↑j : ↥U) ^ (N + 1) • g (↑i : ↥U) ^ N • f (↑i : ↥U) = g (↑i : ↥U) ^ (N + 1) • g (↑j : ↥U) ^ N • f (↑j : ↥U)
have h₁ : g (↑i : ↥U) ^ (N + 1) • g (↑j : ↥U) ^ N • f (↑j : ↥U) = (g (↑i : ↥U) * g (↑j : ↥U)) ^ N • g (↑i : ↥U) • f (↑j : ↥U)
Tactic: convert_to (g i * g ↑j) ^ N • g j • f i = (g i * g ↑j) ^ N • g i • f j