ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- StarAlgEquiv.restrictScalars_injective
theorem StarAlgEquiv.restrictScalars_injective :
    Function.Injective (StarAlgEquiv.restrictScalars R : (A ≃⋆ₐ[S] B) → A ≃⋆ₐ[R] B) 
Arguments: (R : Type u_1) {S : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R S] [Algebra S A] [Algebra S B] [Algebra R A] [Algebra R B] [IsScalarTower R S A] [IsScalarTower R S B] [Star A] [Star B]
Goal: Function.Injective (StarAlgEquiv.restrictScalars R)
-- SheafOfModules.restrictScalars_map_val
@[simps]
noncomputable def restrictScalars :
    SheafOfModules.{v} R' ⥤ SheafOfModules.{v} R 
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R R' : CategoryTheory.Sheaf J RingCat} (α : R ⟶ R')
Goal: ∀ {X Y : SheafOfModules R'} (φ : X ⟶ Y),
  ((SheafOfModules.restrictScalars α).map φ).val = (PresheafOfModules.restrictScalars α.val).map φ.val
-- SheafOfModules.restrictScalars_obj_val
@[simps]
noncomputable def restrictScalars :
    SheafOfModules.{v} R' ⥤ SheafOfModules.{v} R 
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R R' : CategoryTheory.Sheaf J RingCat} (α : R ⟶ R') (M' : SheafOfModules R')
Goal: ((SheafOfModules.restrictScalars α).obj M').val = (PresheafOfModules.restrictScalars α.val).obj M'.val
-- PresheafOfModules.sheafification_map
@[simps! (config := .lemmasOnly) map]
noncomputable def sheafification : PresheafOfModules.{v} R₀ ⥤ SheafOfModules.{v} R 
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R₀ : Cᵒᵖ ⥤ RingCat} {R : CategoryTheory.Sheaf J RingCat} (α : R₀ ⟶ R.val) [CategoryTheory.Presheaf.IsLocallyInjective J α] [CategoryTheory.Presheaf.IsLocallySurjective J α] [J.WEqualsLocallyBijective AddCommGrp] [CategoryTheory.HasWeakSheafify J AddCommGrp]
Goal: ∀ {X Y : PresheafOfModules R₀} (f : X ⟶ Y),
  (PresheafOfModules.sheafification α).map f =
    PresheafOfModules.sheafifyMap α (CategoryTheory.toSheafify J X.presheaf) (CategoryTheory.toSheafify J Y.presheaf) f
      ((CategoryTheory.presheafToSheaf J AddCommGrp).map f.hom) ⋯
-- PresheafOfModules.sheafificationHomEquiv_hom'
lemma sheafificationHomEquiv_hom'
    {P : PresheafOfModules.{v} R₀} {F : SheafOfModules.{v} R}
    (f : (sheafification α).obj P ⟶ F) :
    (sheafificationHomEquiv α f).hom =
      CategoryTheory.toSheafify J P.presheaf ≫ f.val.hom 
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R₀ : Cᵒᵖ ⥤ RingCat} {R : CategoryTheory.Sheaf J RingCat} (α : R₀ ⟶ R.val) [CategoryTheory.Presheaf.IsLocallyInjective J α] [CategoryTheory.Presheaf.IsLocallySurjective J α] [J.WEqualsLocallyBijective AddCommGrp] [CategoryTheory.HasWeakSheafify J AddCommGrp] {P : PresheafOfModules R₀} {F : SheafOfModules R} (f : (PresheafOfModules.sheafification α).obj P ⟶ F)
Goal: ((PresheafOfModules.sheafificationHomEquiv α) f).hom = CategoryTheory.toSheafify J P.presheaf ≫ f.val.hom
-- PresheafOfModules.sheafificationHomEquiv_hom
lemma sheafificationHomEquiv_hom
    {P : PresheafOfModules.{v} R₀} {F : SheafOfModules.{v} R}
    (f : (sheafification α).obj P ⟶ F) :
    (sheafificationHomEquiv α f).hom =
      (sheafificationAdjunction J AddCommGrp).homEquiv P.presheaf
        ((SheafOfModules.toSheaf _).obj F) ((SheafOfModules.toSheaf _).map f) 
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R₀ : Cᵒᵖ ⥤ RingCat} {R : CategoryTheory.Sheaf J RingCat} (α : R₀ ⟶ R.val) [CategoryTheory.Presheaf.IsLocallyInjective J α] [CategoryTheory.Presheaf.IsLocallySurjective J α] [J.WEqualsLocallyBijective AddCommGrp] [CategoryTheory.HasWeakSheafify J AddCommGrp] {P : PresheafOfModules R₀} {F : SheafOfModules R} (f : (PresheafOfModules.sheafification α).obj P ⟶ F)
Goal: ((PresheafOfModules.sheafificationHomEquiv α) f).hom =
  ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P.presheaf ((SheafOfModules.toSheaf R).obj F))
    ((SheafOfModules.toSheaf R).map f)
-- PresheafOfModules.toSheaf_map_sheafificationHomEquiv_symm
lemma toSheaf_map_sheafificationHomEquiv_symm
    {P : PresheafOfModules.{v} R₀} {F : SheafOfModules.{v} R}
    (g : P ⟶ (restrictScalars α).obj ((SheafOfModules.forget _).obj F)) :
    (SheafOfModules.toSheaf _).map ((sheafificationHomEquiv α).symm g) =
      (((sheafificationAdjunction J AddCommGrp).homEquiv
        P.presheaf ((SheafOfModules.toSheaf R).obj F)).symm g.hom) 
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R₀ : Cᵒᵖ ⥤ RingCat} {R : CategoryTheory.Sheaf J RingCat} (α : R₀ ⟶ R.val) [CategoryTheory.Presheaf.IsLocallyInjective J α] [CategoryTheory.Presheaf.IsLocallySurjective J α] [J.WEqualsLocallyBijective AddCommGrp] [CategoryTheory.HasWeakSheafify J AddCommGrp] {P : PresheafOfModules R₀} {F : SheafOfModules R} (g : P ⟶ (PresheafOfModules.restrictScalars α).obj ((SheafOfModules.forget R).obj F))
Goal: (SheafOfModules.toSheaf R).map ((PresheafOfModules.sheafificationHomEquiv α).symm g) =
  ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P.presheaf ((SheafOfModules.toSheaf R).obj F)).symm
    g.hom
-- PresheafOfModules.sheafificationAdjunction_homEquiv_apply
@[simps! (config := .lemmasOnly) homEquiv_apply]
noncomputable def sheafificationAdjunction :
    sheafification.{v} α ⊣ SheafOfModules.forget R ⋙ restrictScalars α 
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R₀ : Cᵒᵖ ⥤ RingCat} {R : CategoryTheory.Sheaf J RingCat} (α : R₀ ⟶ R.val) [CategoryTheory.Presheaf.IsLocallyInjective J α] [CategoryTheory.Presheaf.IsLocallySurjective J α] [J.WEqualsLocallyBijective AddCommGrp] [CategoryTheory.HasWeakSheafify J AddCommGrp] (X : PresheafOfModules R₀) (Y : SheafOfModules R)
Goal: ∀ (a : PresheafOfModules.sheafify α (CategoryTheory.toSheafify J X.presheaf) ⟶ Y),
  ((PresheafOfModules.sheafificationAdjunction α).homEquiv X Y) a =
    (PresheafOfModules.sheafifyHomEquiv' α (CategoryTheory.toSheafify J X.presheaf) ⋯)
      ((SheafOfModules.fullyFaithfulForget R).homEquiv a)
-- PresheafOfModules.sheafificationAdjunction_unit_app_hom
@[simp]
lemma sheafificationAdjunction_unit_app_hom (M₀ : PresheafOfModules.{v} R₀) :
    ((sheafificationAdjunction α).unit.app M₀).hom = CategoryTheory.toSheafify J M₀.presheaf 
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R₀ : Cᵒᵖ ⥤ RingCat} {R : CategoryTheory.Sheaf J RingCat} (α : R₀ ⟶ R.val) [CategoryTheory.Presheaf.IsLocallyInjective J α] [CategoryTheory.Presheaf.IsLocallySurjective J α] [J.WEqualsLocallyBijective AddCommGrp] [CategoryTheory.HasWeakSheafify J AddCommGrp] (M₀ : PresheafOfModules R₀)
Goal: ((PresheafOfModules.sheafificationAdjunction α).unit.app M₀).hom = CategoryTheory.toSheafify J M₀.presheaf
-- CoalgebraCat.toComonObj_X
@[simps] def toComonObj (X : CoalgebraCat R) : Comon_ (ModuleCat R) 
Arguments: {R : Type u} [CommRing R] (X : CoalgebraCat R)
Goal: X.toComonObj.X = ModuleCat.of R ↑X.toModuleCat
-- CoalgebraCat.toComonObj_counit
@[simps] def toComonObj (X : CoalgebraCat R) : Comon_ (ModuleCat R) 
Arguments: {R : Type u} [CommRing R] (X : CoalgebraCat R)
Goal: X.toComonObj.counit = ModuleCat.ofHom Coalgebra.counit
-- CoalgebraCat.toComonObj_comul
@[simps] def toComonObj (X : CoalgebraCat R) : Comon_ (ModuleCat R) 
Arguments: {R : Type u} [CommRing R] (X : CoalgebraCat R)
Goal: X.toComonObj.comul = ModuleCat.ofHom Coalgebra.comul
-- CoalgebraCat.toComon_map_hom
@[simps]
def toComon : CoalgebraCat R ⥤ Comon_ (ModuleCat R) 
Arguments: (R : Type u) [CommRing R]
Goal: ∀ {X Y : CoalgebraCat R} (f : X ⟶ Y), ((CoalgebraCat.toComon R).map f).hom = ModuleCat.ofHom ↑f.toCoalgHom
-- CoalgebraCat.toComon_obj
@[simps]
def toComon : CoalgebraCat R ⥤ Comon_ (ModuleCat R) 
Arguments: (R : Type u) [CommRing R] (X : CoalgebraCat R)
Goal: (CoalgebraCat.toComon R).obj X = X.toComonObj
-- CoalgebraCat.ofComonObjCoalgebraStruct_comul
@[simps]
instance ofComonObjCoalgebraStruct (X : Comon_ (ModuleCat R)) :
    CoalgebraStruct R X.X 
Arguments: {R : Type u} [CommRing R] (X : Comon_ (ModuleCat R))
Goal: Coalgebra.comul = X.comul
-- CoalgebraCat.ofComonObjCoalgebraStruct_counit
@[simps]
instance ofComonObjCoalgebraStruct (X : Comon_ (ModuleCat R)) :
    CoalgebraStruct R X.X 
Arguments: {R : Type u} [CommRing R] (X : Comon_ (ModuleCat R))
Goal: Coalgebra.counit = X.counit
-- CoalgebraCat.comonEquivalence_functor
@[simps]
noncomputable def comonEquivalence : CoalgebraCat R ≌ Comon_ (ModuleCat R) 
Arguments: (R : Type u) [CommRing R]
Goal: (CoalgebraCat.comonEquivalence R).functor = CoalgebraCat.toComon R
-- CoalgebraCat.comonEquivalence_inverse
@[simps]
noncomputable def comonEquivalence : CoalgebraCat R ≌ Comon_ (ModuleCat R) 
Arguments: (R : Type u) [CommRing R]
Goal: (CoalgebraCat.comonEquivalence R).inverse = CoalgebraCat.ofComon R
-- CoalgebraCat.comonEquivalence_unitIso
@[simps]
noncomputable def comonEquivalence : CoalgebraCat R ≌ Comon_ (ModuleCat R) 
Arguments: (R : Type u) [CommRing R]
Goal: (CoalgebraCat.comonEquivalence R).unitIso =
  CategoryTheory.NatIso.ofComponents
    (fun x => CategoryTheory.Iso.refl ((CategoryTheory.Functor.id (CoalgebraCat R)).obj x)) ⋯
-- CoalgebraCat.comonEquivalence_counitIso
@[simps]
noncomputable def comonEquivalence : CoalgebraCat R ≌ Comon_ (ModuleCat R) 
Arguments: (R : Type u) [CommRing R]
Goal: (CoalgebraCat.comonEquivalence R).counitIso =
  CategoryTheory.NatIso.ofComponents
    (fun x => CategoryTheory.Iso.refl (((CoalgebraCat.ofComon R).comp (CoalgebraCat.toComon R)).obj x)) ⋯
-- CoalgebraCat.MonoidalCategoryAux.tensorObj_comul
theorem tensorObj_comul (K L : CoalgebraCat R) :
    Coalgebra.comul (R := R) (A := (K ⊗ L : CoalgebraCat R))
      = (TensorProduct.tensorTensorTensorComm R K K L L).toLinearMap
      ∘ₗ TensorProduct.map Coalgebra.comul Coalgebra.comul 
Arguments: {R : Type u} [CommRing R] (K L : CoalgebraCat R)
Goal: Coalgebra.comul =
  ↑(TensorProduct.tensorTensorTensorComm R ↑K.toModuleCat ↑K.toModuleCat ↑L.toModuleCat ↑L.toModuleCat) ∘ₗ
    TensorProduct.map Coalgebra.comul Coalgebra.comul
-- CoalgebraCat.MonoidalCategoryAux.tensorHom_toLinearMap
theorem tensorHom_toLinearMap (f : M →ₗc[R] N) (g : P →ₗc[R] Q) :
    (CoalgebraCat.ofHom f ⊗ CoalgebraCat.ofHom g).1.toLinearMap
      = TensorProduct.map f.toLinearMap g.toLinearMap 
Arguments: {R : Type u} [CommRing R] {M N P Q : Type u} [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [AddCommGroup Q] [Module R M] [Module R N] [Module R P] [Module R Q] [Coalgebra R M] [Coalgebra R N] [Coalgebra R P] [Coalgebra R Q] (f : M →ₗc[R] N) (g : P →ₗc[R] Q)
Goal: (CategoryTheory.MonoidalCategory.tensorHom (CoalgebraCat.ofHom f) (CoalgebraCat.ofHom g)).toCoalgHom.toLinearMap =
  TensorProduct.map f.toLinearMap g.toLinearMap
-- CoalgebraCat.MonoidalCategoryAux.associator_hom_toLinearMap
theorem associator_hom_toLinearMap :
    (α_ (CoalgebraCat.of R M) (CoalgebraCat.of R N) (CoalgebraCat.of R P)).hom.1.toLinearMap
      = (TensorProduct.assoc R M N P).toLinearMap 
Arguments: {R : Type u} [CommRing R] {M N P : Type u} [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [Coalgebra R M] [Coalgebra R N] [Coalgebra R P]
Goal: (CategoryTheory.MonoidalCategory.associator (CoalgebraCat.of R M) (CoalgebraCat.of R N)
          (CoalgebraCat.of R P)).hom.toCoalgHom.toLinearMap =
  ↑(TensorProduct.assoc R M N P)
-- CoalgebraCat.MonoidalCategoryAux.leftUnitor_hom_toLinearMap
theorem leftUnitor_hom_toLinearMap :
    (λ_ (CoalgebraCat.of R M)).hom.1.toLinearMap = (TensorProduct.lid R M).toLinearMap 
Arguments: {R : Type u} [CommRing R] {M : Type u} [AddCommGroup M] [Module R M] [Coalgebra R M]
Goal: (CategoryTheory.MonoidalCategory.leftUnitor (CoalgebraCat.of R M)).hom.toCoalgHom.toLinearMap = ↑(TensorProduct.lid R M)
-- CoalgebraCat.MonoidalCategoryAux.rightUnitor_hom_toLinearMap
theorem rightUnitor_hom_toLinearMap :
    (ρ_ (CoalgebraCat.of R M)).hom.1.toLinearMap = (TensorProduct.rid R M).toLinearMap 
Arguments: {R : Type u} [CommRing R] {M : Type u} [AddCommGroup M] [Module R M] [Coalgebra R M]
Goal: (CategoryTheory.MonoidalCategory.rightUnitor (CoalgebraCat.of R M)).hom.toCoalgHom.toLinearMap =
  ↑(TensorProduct.rid R M)
-- CoalgebraCat.MonoidalCategoryAux.comul_tensorObj
theorem comul_tensorObj :
    Coalgebra.comul (R := R) (A := (CoalgebraCat.of R M ⊗ CoalgebraCat.of R N : CoalgebraCat R))
      = Coalgebra.comul (A := M ⊗[R] N) 
Arguments: {R : Type u} [CommRing R] {M N : Type u} [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [Coalgebra R M] [Coalgebra R N]
Goal: Coalgebra.comul = Coalgebra.comul
-- CoalgebraCat.MonoidalCategoryAux.comul_tensorObj_tensorObj_right
theorem comul_tensorObj_tensorObj_right :
    Coalgebra.comul (R := R) (A := (CoalgebraCat.of R M ⊗
      (CoalgebraCat.of R N ⊗ CoalgebraCat.of R P) : CoalgebraCat R))
      = Coalgebra.comul (A := M ⊗[R] N ⊗[R] P) 
Arguments: {R : Type u} [CommRing R] {M N P : Type u} [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [Coalgebra R M] [Coalgebra R N] [Coalgebra R P]
Goal: Coalgebra.comul = Coalgebra.comul
-- CoalgebraCat.MonoidalCategoryAux.comul_tensorObj_tensorObj_left
theorem comul_tensorObj_tensorObj_left :
    Coalgebra.comul (R := R)
      (A := ((CoalgebraCat.of R M ⊗ CoalgebraCat.of R N) ⊗ CoalgebraCat.of R P : CoalgebraCat R))
      = Coalgebra.comul (A := (M ⊗[R] N) ⊗[R] P) 
Arguments: {R : Type u} [CommRing R] {M N P : Type u} [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [Coalgebra R M] [Coalgebra R N] [Coalgebra R P]
Goal: Coalgebra.comul = Coalgebra.comul
-- CoalgebraCat.MonoidalCategoryAux.counit_tensorObj
theorem counit_tensorObj :
    Coalgebra.counit (R := R) (A := (CoalgebraCat.of R M ⊗ CoalgebraCat.of R N : CoalgebraCat R))
      = Coalgebra.counit (A := M ⊗[R] N) 
Arguments: {R : Type u} [CommRing R] {M N : Type u} [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [Coalgebra R M] [Coalgebra R N]
Goal: Coalgebra.counit = Coalgebra.counit
-- CoalgebraCat.MonoidalCategoryAux.counit_tensorObj_tensorObj_right
theorem counit_tensorObj_tensorObj_right :
    Coalgebra.counit (R := R)
      (A := (CoalgebraCat.of R M ⊗ (CoalgebraCat.of R N ⊗ CoalgebraCat.of R P) : CoalgebraCat R))
      = Coalgebra.counit (A := M ⊗[R] (N ⊗[R] P)) 
Arguments: {R : Type u} [CommRing R] {M N P : Type u} [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [Coalgebra R M] [Coalgebra R N] [Coalgebra R P]
Goal: Coalgebra.counit = Coalgebra.counit
-- CoalgebraCat.MonoidalCategoryAux.counit_tensorObj_tensorObj_left
theorem counit_tensorObj_tensorObj_left :
    Coalgebra.counit (R := R)
      (A := ((CoalgebraCat.of R M ⊗ CoalgebraCat.of R N) ⊗ CoalgebraCat.of R P : CoalgebraCat R))
      = Coalgebra.counit (A := (M ⊗[R] N) ⊗[R] P) 
Arguments: {R : Type u} [CommRing R] {M N P : Type u} [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [Coalgebra R M] [Coalgebra R N] [Coalgebra R P]
Goal: Coalgebra.counit = Coalgebra.counit
-- SheafOfModules.GeneratingSections.epi
structure GeneratingSections where
    I : Type u
    s : I → M.sections
  epi : Epi (M.freeHomEquiv.symm s) := by infer_instance
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R : CategoryTheory.Sheaf J RingCat} [CategoryTheory.HasWeakSheafify J AddCommGrp] [J.WEqualsLocallyBijective AddCommGrp] [J.HasSheafCompose (CategoryTheory.forget₂ RingCat AddCommGrp)] {M : SheafOfModules R} (self : M.GeneratingSections)
Goal: CategoryTheory.Epi (M.freeHomEquiv.symm self.s)
-- SheafOfModules.GeneratingSections.ofEpi_s
@[simps]
def ofEpi (σ : M.GeneratingSections) (p : M ⟶ N) [Epi p] :
    N.GeneratingSections 
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R : CategoryTheory.Sheaf J RingCat} [CategoryTheory.HasWeakSheafify J AddCommGrp] [J.WEqualsLocallyBijective AddCommGrp] [J.HasSheafCompose (CategoryTheory.forget₂ RingCat AddCommGrp)] {M N : SheafOfModules R} (σ : M.GeneratingSections) (p : M ⟶ N) [CategoryTheory.Epi p] (i : σ.I)
Goal: (σ.ofEpi p).s i = SheafOfModules.sectionsMap p (σ.s i)
-- SheafOfModules.GeneratingSections.ofEpi_I
@[simps]
def ofEpi (σ : M.GeneratingSections) (p : M ⟶ N) [Epi p] :
    N.GeneratingSections 
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R : CategoryTheory.Sheaf J RingCat} [CategoryTheory.HasWeakSheafify J AddCommGrp] [J.WEqualsLocallyBijective AddCommGrp] [J.HasSheafCompose (CategoryTheory.forget₂ RingCat AddCommGrp)] {M N : SheafOfModules R} (σ : M.GeneratingSections) (p : M ⟶ N) [CategoryTheory.Epi p]
Goal: (σ.ofEpi p).I = σ.I
-- SheafOfModules.GeneratingSections.opEpi_id
lemma opEpi_id (σ : M.GeneratingSections) :
    σ.ofEpi (𝟙 M) = σ 
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R : CategoryTheory.Sheaf J RingCat} [CategoryTheory.HasWeakSheafify J AddCommGrp] [J.WEqualsLocallyBijective AddCommGrp] [J.HasSheafCompose (CategoryTheory.forget₂ RingCat AddCommGrp)] {M : SheafOfModules R} (σ : M.GeneratingSections)
Goal: σ.ofEpi (𝟙 M) = σ
-- SheafOfModules.GeneratingSections.opEpi_comp
lemma opEpi_comp (σ : M.GeneratingSections) (p : M ⟶ N) (q : N ⟶ P) [Epi p] [Epi q] :
    σ.ofEpi (p ≫ q) = (σ.ofEpi p).ofEpi q 
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R : CategoryTheory.Sheaf J RingCat} [CategoryTheory.HasWeakSheafify J AddCommGrp] [J.WEqualsLocallyBijective AddCommGrp] [J.HasSheafCompose (CategoryTheory.forget₂ RingCat AddCommGrp)] {M N P : SheafOfModules R} (σ : M.GeneratingSections) (p : M ⟶ N) (q : N ⟶ P) [CategoryTheory.Epi p] [CategoryTheory.Epi q]
Goal: σ.ofEpi (p ≫ q) = (σ.ofEpi p).ofEpi q
-- SheafOfModules.LocalGeneratorsData.coversTop
structure LocalGeneratorsData where
    I : Type u'
    X : I → C
  coversTop : J.CoversTop X
    generators (i : I) : (M.over (X i)).GeneratingSections
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R : CategoryTheory.Sheaf J RingCat} {M : SheafOfModules R} [∀ (X : C), CategoryTheory.HasWeakSheafify (J.over X) AddCommGrp] [∀ (X : C), (J.over X).WEqualsLocallyBijective AddCommGrp] [∀ (X : C), (J.over X).HasSheafCompose (CategoryTheory.forget₂ RingCat AddCommGrp)] (self : M.LocalGeneratorsData)
Goal: J.CoversTop self.X
-- SheafOfModules.IsFiniteType.exists_localGeneratorsData
class IsFiniteType : Prop where
  exists_localGeneratorsData :
    ∃ (σ : M.LocalGeneratorsData), ∀ (i : σ.I), Finite (σ.generators i).I
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R : CategoryTheory.Sheaf J RingCat} {M : SheafOfModules R} [∀ (X : C), CategoryTheory.HasWeakSheafify (J.over X) AddCommGrp] [∀ (X : C), (J.over X).WEqualsLocallyBijective AddCommGrp] [∀ (X : C), (J.over X).HasSheafCompose (CategoryTheory.forget₂ RingCat AddCommGrp)] [self : M.IsFiniteType]
Goal: ∃ σ, ∀ (i : σ.I), Finite (σ.generators i).I
-- SheafOfModules.pushforward_map_val
@[simps]
noncomputable def pushforward : SheafOfModules.{v} R ⥤ SheafOfModules.{v} S 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {J : CategoryTheory.GrothendieckTopology C} {K : CategoryTheory.GrothendieckTopology D} {F : C ⥤ D} {S : CategoryTheory.Sheaf J RingCat} {R : CategoryTheory.Sheaf K RingCat} [F.IsContinuous J K] [F.IsContinuous J K] (φ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R)
Goal: ∀ {X Y : SheafOfModules R} (f : X ⟶ Y),
  ((SheafOfModules.pushforward φ).map f).val = (PresheafOfModules.pushforward φ.val).map f.val
-- SheafOfModules.pushforward_obj_val
@[simps]
noncomputable def pushforward : SheafOfModules.{v} R ⥤ SheafOfModules.{v} S 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {J : CategoryTheory.GrothendieckTopology C} {K : CategoryTheory.GrothendieckTopology D} {F : C ⥤ D} {S : CategoryTheory.Sheaf J RingCat} {R : CategoryTheory.Sheaf K RingCat} [F.IsContinuous J K] [F.IsContinuous J K] (φ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R) (M : SheafOfModules R)
Goal: ((SheafOfModules.pushforward φ).obj M).val = (PresheafOfModules.pushforward φ.val).obj M.val
-- SheafOfModules.freeHomEquiv_comp_apply
lemma freeHomEquiv_comp_apply {M N : SheafOfModules.{u} R} {I : Type u}
    (f : free I ⟶ M) (p : M ⟶ N) (i : I) :
    N.freeHomEquiv (f ≫ p) i = sectionsMap p (M.freeHomEquiv f i) 
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R : CategoryTheory.Sheaf J RingCat} [CategoryTheory.HasWeakSheafify J AddCommGrp] [J.WEqualsLocallyBijective AddCommGrp] [J.HasSheafCompose (CategoryTheory.forget₂ RingCat AddCommGrp)] {M N : SheafOfModules R} {I : Type u} (f : SheafOfModules.free I ⟶ M) (p : M ⟶ N) (i : I)
Goal: N.freeHomEquiv (f ≫ p) i = SheafOfModules.sectionsMap p (M.freeHomEquiv f i)
-- SheafOfModules.freeHomEquiv_symm_comp
lemma freeHomEquiv_symm_comp {M N : SheafOfModules.{u} R} {I : Type u} (s : I → M.sections)
    (p : M ⟶ N) :
    M.freeHomEquiv.symm s ≫ p = N.freeHomEquiv.symm (fun i ↦ sectionsMap p (s i)) 
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R : CategoryTheory.Sheaf J RingCat} [CategoryTheory.HasWeakSheafify J AddCommGrp] [J.WEqualsLocallyBijective AddCommGrp] [J.HasSheafCompose (CategoryTheory.forget₂ RingCat AddCommGrp)] {M N : SheafOfModules R} {I : Type u} (s : I → M.sections) (p : M ⟶ N)
Goal: M.freeHomEquiv.symm s ≫ p = N.freeHomEquiv.symm fun i => SheafOfModules.sectionsMap p (s i)
-- SheafOfModules.freeHomEquiv_freeMap
@[simp]
lemma freeHomEquiv_freeMap :
    (freeHomEquiv _ (freeMap (R := R) f)) = freeSection.comp f 
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R : CategoryTheory.Sheaf J RingCat} [CategoryTheory.HasWeakSheafify J AddCommGrp] [J.WEqualsLocallyBijective AddCommGrp] [J.HasSheafCompose (CategoryTheory.forget₂ RingCat AddCommGrp)] {I : Type u}
Goal: ∀ {J_1 : Type u} (f : I → J_1),
  (SheafOfModules.free J_1).freeHomEquiv (SheafOfModules.freeMap f) = SheafOfModules.freeSection ∘ f
-- SheafOfModules.sectionMap_freeMap_freeSection
@[simp]
lemma sectionMap_freeMap_freeSection (i : I) :
    sectionsMap (freeMap (R := R) f) (freeSection i) = freeSection (f i) 
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R : CategoryTheory.Sheaf J RingCat} [CategoryTheory.HasWeakSheafify J AddCommGrp] [J.WEqualsLocallyBijective AddCommGrp] [J.HasSheafCompose (CategoryTheory.forget₂ RingCat AddCommGrp)] {I : Type u}
Goal: ∀ {J_1 : Type u} (f : I → J_1) (i : I),
  SheafOfModules.sectionsMap (SheafOfModules.freeMap f) (SheafOfModules.freeSection i) =
    SheafOfModules.freeSection (f i)
-- SheafOfModules.QuasicoherentData.coversTop
structure QuasicoherentData where
    I : Type u'
    X : I → C
  coversTop : J.CoversTop X
    presentation (i : I) : (M.over (X i)).Presentation
Arguments: {C : Type u'} [CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C} {R : CategoryTheory.Sheaf J RingCat} {M : SheafOfModules R} [∀ (X : C), (J.over X).HasSheafCompose (CategoryTheory.forget₂ RingCat AddCommGrp)] [∀ (X : C), CategoryTheory.HasWeakSheafify (J.over X) AddCommGrp] [∀ (X : C), (J.over X).WEqualsLocallyBijective AddCommGrp] (self : M.QuasicoherentData)
Goal: J.CoversTop self.X