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
this :
  IsLocalizedModule (M' :=
    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 ⊤))
    ⊥ (toOpenₗ R M ⊤)
x :
  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 ⊤)
⊢ ∃ (a : M), (toOpenₗ R M ⊤ : (a : M) → (structureSheafInType R M).obj.obj (op ⊤)) a = x
have obtain : ∀ (x : M),
  ∃ (a : M),
    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 ⊤))
      ((toOpenₗ R M ⊤ : (a : M) → (structureSheafInType R M).obj.obj (op ⊤)) a :
        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 ⊤))
      (Function.uncurry (β := ↥⊥) (IsLocalizedModule.mk' (toOpenₗ R M ⊤))
          (x, Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submonoid R) ⊥ x) 1 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 ⊤))
Tactic: obtain ⟨⟨x, _, rfl⟩, rfl⟩ := IsLocalizedModule.mk'_surjective ⊥ (toOpenₗ R M ⊤) x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (goal)
Goal:
R : Type u
inst : CommRing R
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
x : (↑(PrimeSpectrum.Top R) : Type u)
hxU : x ∈ U
⊢ Eq (α := CommRingCat.of R ⟶ Presheaf.stalk (structurePresheafInCommRingCat R) x)
    (CommRingCat.ofHom
        (algebraMap R
          (Functor.obj
            (ObjectProperty.FullSubcategory.obj (P :=
              CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
              (structureSheafInType R R))
            (op U))) ≫
      Presheaf.germ (structurePresheafInCommRingCat R) U x hxU)
    (CategoryStruct.comp (Y :=
      CommRingCat.of
        (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 R))
          (op ⊤)))
      (CommRingCat.ofHom
        (algebraMap R
          (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 R))
            (op ⊤))))
      (Presheaf.germ (structurePresheafInCommRingCat R) ⊤ x _root_.trivial))
have rw : Eq (α := CommRingCat.of R ⟶ Presheaf.stalk (structurePresheafInCommRingCat R) x)
  (CommRingCat.ofHom
      (algebraMap R
        (Functor.obj
          (ObjectProperty.FullSubcategory.obj (P :=
            CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
            (structureSheafInType R R))
          (op U))) ≫
    Presheaf.germ (structurePresheafInCommRingCat R) U x hxU)
  (CategoryStruct.comp (Y :=
    CommRingCat.of
      (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 R))
        (op ⊤)))
    (CommRingCat.ofHom
      (algebraMap R
        (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 R))
          (op ⊤))))
    (CategoryTheory.Functor.map (structurePresheafInCommRingCat R) (Quiver.Hom.op (homOfLE sorry)) ≫
      Presheaf.germ (structurePresheafInCommRingCat R) U x hxU))
Tactic: rw [← (structurePresheafInCommRingCat R).germ_res (homOfLE (le_top : U ≤ ⊤)) _ hxU]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (goal)
Goal:
R M A : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : CommRing A
inst_4 : Algebra R A
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
x : (↑(PrimeSpectrum.Top R) : Type u)
hxU : x ∈ U
r : R
m :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
⊢ Eq (α := (↑(Presheaf.stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u))
    (ZeroHom.toFun
      (↑(AddCommGrpCat.Hom.hom (Presheaf.germ (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) U x hxU)) :
        ZeroHom (↑((moduleStructurePresheaf R M).presheaf.obj (op U)) : Type u)
          (↑(Presheaf.stalk (moduleStructurePresheaf R M).presheaf x) : Type u))
      (r • m))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
          AddCommGrpCat.carrier) (X := Functor.obj (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) (op U))
          (Y := Presheaf.stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x)
          (Presheaf.germ (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) U x hxU) :
        (a : (↑((moduleStructurePresheaf R M).presheaf.obj (op U)) : Type u)) →
          (↑(Presheaf.stalk (moduleStructurePresheaf R M).presheaf x) : Type u))
      (HSMul.hSMul (α := (↑(Functor.obj (structurePresheafInCommRingCat R) (op U)) : Type u))
        ((algebraMap R
                (Functor.obj
                  (ObjectProperty.FullSubcategory.obj (P :=
                    CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                    (structureSheafInType R R))
                  (op U)) :
              (a : R) → (structureSheafInType R R).obj.obj (op U))
            ((RingHom.id R : (a : R) → R) r) :
          Functor.obj
            (ObjectProperty.FullSubcategory.obj (P :=
              CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
              (structureSheafInType R R))
            (op U))
        m))
have e_a : r • m =
  HSMul.hSMul (α := (↑(Functor.obj (structurePresheafInCommRingCat R) (op U)) : Type u))
    ((algebraMap R
            (Functor.obj
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R R))
              (op U)) :
          (a : R) → (structureSheafInType R R).obj.obj (op U))
        ((RingHom.id R : (a : R) → R) r) :
      Functor.obj
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R R))
        (op U))
    m
Tactic: congr 1
-- 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
x : (↑(PrimeSpectrum.Top R) : Type u)
r : R
m : (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u)
⊢ (modulePresheafStalkIso R M x :
        (a : (↑(Presheaf.stalk (moduleStructurePresheaf R M).presheaf x) : Type u)) →
          (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
      (HSMul.hSMul (α := (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u)) (β :=
        (↑(Presheaf.stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (toStalk R x) :
              (a : (↑(CommRingCat.of R) : Type u)) → (↑((structurePresheafInCommRingCat R).stalk x) : Type u))
            r :
          (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u))
        ((LinearEquiv.symm (modulePresheafStalkIso R M x) :
              (a : (↑((structurePresheafInModuleCat R M).stalk x) : Type u)) →
                (↑(Presheaf.stalk (moduleStructurePresheaf R M).presheaf x) : Type u))
            m :
          (↑(Presheaf.stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u))) =
    r • m
have rw : HSMul.hSMul (α := (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u)) (β :=
    (↑(Presheaf.stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (toStalk R x) :
          (a : (↑(CommRingCat.of R) : Type u)) → (↑((structurePresheafInCommRingCat R).stalk x) : Type u))
        r :
      (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u))
    ((LinearEquiv.symm (modulePresheafStalkIso R M x) :
          (a : (↑((structurePresheafInModuleCat R M).stalk x) : Type u)) →
            (↑(Presheaf.stalk (moduleStructurePresheaf R M).presheaf x) : Type u))
        m :
      (↑(Presheaf.stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u)) =
  HSMul.hSMul (β := (↑(Presheaf.stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u)) r
    ((LinearEquiv.symm (modulePresheafStalkIso R M x) :
          (a : (↑((structurePresheafInModuleCat R M).stalk x) : Type u)) →
            (↑(Presheaf.stalk (moduleStructurePresheaf R M).presheaf x) : Type u))
        m :
      (↑(Presheaf.stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u))
Tactic: rw [← (modulePresheafStalkIso R M x).eq_symm_apply, map_smul]
-- 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)
x : (↑(PrimeSpectrum.Top R) : Type u)
hx : x ∈ U
⊢ Eq (α := ModuleCat.of R M ⟶ Presheaf.stalk (structurePresheafInModuleCat R M) x)
    (ModuleCat.ofHom (toOpenₗ R M U) ≫ Presheaf.germ (structurePresheafInModuleCat R M) U x hx) (toStalkₗ' R M x)
have rw : Eq (α := ModuleCat.of R M ⟶ Presheaf.stalk (structurePresheafInModuleCat R M) x)
  (ModuleCat.ofHom (toOpenₗ R M U) ≫ Presheaf.germ (structurePresheafInModuleCat R M) U x hx)
  (CategoryStruct.comp (Y :=
      ModuleCat.of R
        (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 ⊤)))
      (ModuleCat.ofHom (toOpenₗ R M ⊤))
      (CategoryTheory.Functor.map (structurePresheafInModuleCat R M) (Quiver.Hom.op (homOfLE sorry))) ≫
    Presheaf.germ (structurePresheafInModuleCat R M) U x hx)
Tactic: rw [toStalkₗ', ← Presheaf.germ_res _ (homOfLE le_top) _ hx, ← Category.assoc]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (goal)
Goal:
R : Type u
inst : CommRing R
x : (↑(PrimeSpectrum.Top R) : Type u)
f : R
hf : Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen f) x
⊢ IsUnit (M := (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (toStalk R x) :
          (a : (↑(CommRingCat.of R) : Type u)) → (↑((structurePresheafInCommRingCat R).stalk x) : Type u))
        f :
      (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u))
have h : Eq (α := (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u))
  ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
          CommRingCat.carrier) (toStalk R x) :
        (a : (↑(CommRingCat.of R) : Type u)) → (↑((structurePresheafInCommRingCat R).stalk x) : Type u))
      f :
    (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u))
  ((CommRingCat.Hom.hom (Presheaf.germ (structurePresheafInCommRingCat R) (basicOpen f) x hf) :
        (a : (structureSheafInType R R).obj.obj (op (basicOpen f))) →
          (↑((structurePresheafInCommRingCat R).stalk x) : Type u))
      ((algebraMap R
              (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 R))
                (op (basicOpen f))) :
            (a : R) → (structureSheafInType R R).obj.obj (op (basicOpen f)))
          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 R))
          (op (basicOpen f))) :
    (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u))
Tactic: convert! (isUnit_basicOpen f).map ((structurePresheafInCommRingCat R).germ _ x hf).hom
-- 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
x : (↑(PrimeSpectrum.Top R) : Type u)
f : R
hf : Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen f) x
⊢ IsUnit (M := Module.End R (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u))
    ((algebraMap R (Module.End R (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u)) :
          (a : R) → Module.End R (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        f :
      Module.End R (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u))
have this : IsUnit (M :=
  Module.End (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u)
    (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u))
  ((algebraMap (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u)
          (Module.End (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u)
            (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u)) :
        (a : (↑((structurePresheafInCommRingCat R).stalk x) : Type u)) →
          Module.End (↑((structurePresheafInCommRingCat R).stalk x) : Type u)
            (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (toStalk R x) :
          (a : (↑(CommRingCat.of R) : Type u)) → (↑((structurePresheafInCommRingCat R).stalk x) : Type u))
        f) :
    Module.End (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u)
      (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u))
Tactic: have := (isUnit_toStalk x f hf).map (algebraMap _
    (Module.End ((structurePresheafInCommRingCat R).stalk x)
      ((structurePresheafInModuleCat R M).stalk x)))
-- 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
x : (↑(PrimeSpectrum.Top R) : Type u)
f : R
hf : Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen f) x
this :
  Function.Bijective (α := (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u)) (β :=
    (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u))
    (⇑((algebraMap (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u)
                (Module.End (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u)
                  (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u)) :
              (a : (↑((structurePresheafInCommRingCat R).stalk x) : Type u)) →
                Module.End (↑((structurePresheafInCommRingCat R).stalk x) : Type u)
                  (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
            ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                  CommRingCat.carrier) (toStalk R x) :
                (a : (↑(CommRingCat.of R) : Type u)) → (↑((structurePresheafInCommRingCat R).stalk x) : Type u))
              f) :
          Module.End (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u)
            (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u)) :
      (a : (↑((structurePresheafInModuleCat R M).stalk x) : Type u)) →
        (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
a : (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u)
⊢ f • a =
    HSMul.hSMul (α := (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (toStalk R x) :
            (a : R) → (↑((structurePresheafInCommRingCat R).stalk x) : Type u))
          f :
        (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u))
      a
have h : f • a = f • a
Tactic: rw [toStalk_smul]
-- 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
x : (↑(PrimeSpectrum.Top R) : Type u)
f : M
s : ↥x.asIdeal.primeCompl
⊢ HSMul.hSMul (β := (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u)) (↑s : R)
      ((IsLocalizedModule.lift x.asIdeal.primeCompl (LocalizedModule.mkLinearMap x.asIdeal.primeCompl M)
              (ModuleCat.Hom.hom (toStalkₗ' R M x)) ⋯ :
            (a : LocalizedModule x.asIdeal.primeCompl M) → (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
          (LocalizedModule.mk f s) :
        (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u)) =
    HSMul.hSMul (β := (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u)) (↑s : R)
      ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u)) (CC :=
              ModuleCat.carrier) (X := Functor.obj (structurePresheafInModuleCat R M) (op (basicOpen (↑s : R)))) (Y :=
              Presheaf.stalk (structurePresheafInModuleCat R M) x)
              (Presheaf.germ (structurePresheafInModuleCat R M) (basicOpen (↑s : R)) x ⋯) :
            (a : (structureSheafInType R M).obj.obj (op (basicOpen (↑s : R)))) →
              (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
          (const f (↑s : R) (basicOpen (↑s : R)) ⋯) :
        (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u))
have rw : Eq (α := (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u))
  ((ModuleCat.Hom.hom
          (CategoryStruct.comp (Y :=
            ModuleCat.of R
              (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 : R)))))
            (ModuleCat.ofHom (toOpenₗ R M (basicOpen (↑s : R))))
            (Presheaf.germ (structurePresheafInModuleCat R M) (basicOpen (↑s : R)) x sorry)) :
        (a : M) → (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
      f :
    (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u))
  ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u)) (CC :=
          ModuleCat.carrier) (X := Functor.obj (structurePresheafInModuleCat R M) (op (basicOpen (↑s : R)))) (Y :=
          Presheaf.stalk (structurePresheafInModuleCat R M) x)
          (Presheaf.germ (structurePresheafInModuleCat R M) (basicOpen (↑s : R)) x sorry) :
        (a : (structureSheafInType R M).obj.obj (op (basicOpen (↑s : R)))) →
          (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
      (const ((↑s : R) • f) (↑s : R) (basicOpen (↑s : R)) sorry) :
    (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u))
Tactic: rw [← map_smul, LocalizedModule.smul'_mk, ← Submonoid.smul_def, LocalizedModule.mk_cancel,
    ← LocalizedModule.mkLinearMap_apply, IsLocalizedModule.lift_apply, ← map_smul,
    ← toOpenₗ_germ (basicOpen ↑s) _ s.2, 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
x : (↑(PrimeSpectrum.Top R) : Type u)
f : M
s : ↥x.asIdeal.primeCompl
⊢ Eq (α := (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u))
    (((ModuleCat.Hom.hom (Presheaf.germ (structurePresheafInModuleCat R M) (basicOpen (↑s : R)) x ⋯) :
            (↑(ModuleCat.of R
                    (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 : R))))) :
                Type u) →ₗ[R]
              (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u)) :
          (a : (structureSheafInType R M).obj.obj (op (basicOpen (↑s : R)))) →
            (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        (const f 1 (basicOpen (↑s : R)) ⋯) :
      (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (X := Functor.obj (structurePresheafInModuleCat R M) (op (basicOpen (↑s : R)))) (Y :=
            Presheaf.stalk (structurePresheafInModuleCat R M) x)
            (Presheaf.germ (structurePresheafInModuleCat R M) (basicOpen (↑s : R)) x ⋯) :
          (a : (structureSheafInType R M).obj.obj (op (basicOpen (↑s : R)))) →
            (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        (const ((↑s : R) • f) (↑s : R) (basicOpen (↑s : R)) ⋯) :
      (↑(Presheaf.stalk (structurePresheafInModuleCat R M) x) : Type u))
have h : 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 : R))))
  (const f 1 (basicOpen (↑s : R)) sorry) (const ((↑s : R) • f) (↑s : R) (basicOpen (↑s : R)) sorry)
Tactic: congr 1