ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- NonUnitalStarSubalgebra.mem_center_iff
theorem mem_center_iff {a : A} : a ∈ center R A ↔ ∀ b : A, b * a = a * b 
Arguments: {R : Type u} {A : Type v} [CommSemiring R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {a : A}
Goal: a ∈ NonUnitalStarSubalgebra.center R A ↔ ∀ (b : A), b * a = a * b
-- NonUnitalStarSubalgebra.coe_centralizer
@[simp, norm_cast]
theorem coe_centralizer (s : Set A) : (centralizer R s : Set A) = (s ∪ star s).centralizer 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] (s : Set A)
Goal: ↑(NonUnitalStarSubalgebra.centralizer R s) = (s ∪ star s).centralizer
-- NonUnitalStarSubalgebra.mem_centralizer_iff
theorem mem_centralizer_iff {s : Set A} {z : A} :
    z ∈ centralizer R s ↔ ∀ g ∈ s, g * z = z * g ∧ star g * z = z * star g 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] {s : Set A} {z : A}
Goal: z ∈ NonUnitalStarSubalgebra.centralizer R s ↔ ∀ g ∈ s, g * z = z * g ∧ star g * z = z * star g
-- NonUnitalStarSubalgebra.centralizer_le
theorem centralizer_le (s t : Set A) (h : s ⊆ t) : centralizer R t ≤ centralizer R s 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [StarRing R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [StarModule R A] (s t : Set A) (h : s ⊆ t)
Goal: NonUnitalStarSubalgebra.centralizer R t ≤ NonUnitalStarSubalgebra.centralizer R s
-- NonUnitalStarSubalgebra.centralizer_univ
@[simp]
theorem centralizer_univ : centralizer R Set.univ = center R A 
Arguments: (R : Type u) {A : Type v} [CommSemiring R] [NonUnitalSemiring A] [StarRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
Goal: NonUnitalStarSubalgebra.centralizer R Set.univ = NonUnitalStarSubalgebra.center R A
-- Quaternion.re_fst_dualNumberEquiv
@[simp]
theorem re_fst_dualNumberEquiv (q : Quaternion (DualNumber R)) :
    (dualNumberEquiv q).fst.re = q.re.fst 
Arguments: {R : Type u_1} [CommRing R] (q : Quaternion (DualNumber R))
Goal: (TrivSqZeroExt.fst (Quaternion.dualNumberEquiv q)).re = TrivSqZeroExt.fst q.re
-- Quaternion.imI_fst_dualNumberEquiv
@[simp]
theorem imI_fst_dualNumberEquiv (q : Quaternion (DualNumber R)) :
    (dualNumberEquiv q).fst.imI = q.imI.fst 
Arguments: {R : Type u_1} [CommRing R] (q : Quaternion (DualNumber R))
Goal: (TrivSqZeroExt.fst (Quaternion.dualNumberEquiv q)).imI = TrivSqZeroExt.fst q.imI
-- Quaternion.imJ_fst_dualNumberEquiv
@[simp]
theorem imJ_fst_dualNumberEquiv (q : Quaternion (DualNumber R)) :
    (dualNumberEquiv q).fst.imJ = q.imJ.fst 
Arguments: {R : Type u_1} [CommRing R] (q : Quaternion (DualNumber R))
Goal: (TrivSqZeroExt.fst (Quaternion.dualNumberEquiv q)).imJ = TrivSqZeroExt.fst q.imJ
-- Quaternion.imK_fst_dualNumberEquiv
@[simp]
theorem imK_fst_dualNumberEquiv (q : Quaternion (DualNumber R)) :
    (dualNumberEquiv q).fst.imK = q.imK.fst 
Arguments: {R : Type u_1} [CommRing R] (q : Quaternion (DualNumber R))
Goal: (TrivSqZeroExt.fst (Quaternion.dualNumberEquiv q)).imK = TrivSqZeroExt.fst q.imK
-- Quaternion.re_snd_dualNumberEquiv
@[simp]
theorem re_snd_dualNumberEquiv (q : Quaternion (DualNumber R)) :
    (dualNumberEquiv q).snd.re = q.re.snd 
Arguments: {R : Type u_1} [CommRing R] (q : Quaternion (DualNumber R))
Goal: (TrivSqZeroExt.snd (Quaternion.dualNumberEquiv q)).re = TrivSqZeroExt.snd q.re
-- Quaternion.imI_snd_dualNumberEquiv
@[simp]
theorem imI_snd_dualNumberEquiv (q : Quaternion (DualNumber R)) :
    (dualNumberEquiv q).snd.imI = q.imI.snd 
Arguments: {R : Type u_1} [CommRing R] (q : Quaternion (DualNumber R))
Goal: (TrivSqZeroExt.snd (Quaternion.dualNumberEquiv q)).imI = TrivSqZeroExt.snd q.imI
-- Quaternion.imJ_snd_dualNumberEquiv
@[simp]
theorem imJ_snd_dualNumberEquiv (q : Quaternion (DualNumber R)) :
    (dualNumberEquiv q).snd.imJ = q.imJ.snd 
Arguments: {R : Type u_1} [CommRing R] (q : Quaternion (DualNumber R))
Goal: (TrivSqZeroExt.snd (Quaternion.dualNumberEquiv q)).imJ = TrivSqZeroExt.snd q.imJ
-- Quaternion.imK_snd_dualNumberEquiv
@[simp]
theorem imK_snd_dualNumberEquiv (q : Quaternion (DualNumber R)) :
    (dualNumberEquiv q).snd.imK = q.imK.snd 
Arguments: {R : Type u_1} [CommRing R] (q : Quaternion (DualNumber R))
Goal: (TrivSqZeroExt.snd (Quaternion.dualNumberEquiv q)).imK = TrivSqZeroExt.snd q.imK
-- Quaternion.fst_re_dualNumberEquiv_symm
@[simp]
theorem fst_re_dualNumberEquiv_symm (d : DualNumber (Quaternion R)) :
    (dualNumberEquiv.symm d).re.fst = d.fst.re 
Arguments: {R : Type u_1} [CommRing R] (d : DualNumber (Quaternion R))
Goal: TrivSqZeroExt.fst (Quaternion.dualNumberEquiv.symm d).re = (TrivSqZeroExt.fst d).re
-- Quaternion.fst_imI_dualNumberEquiv_symm
@[simp]
theorem fst_imI_dualNumberEquiv_symm (d : DualNumber (Quaternion R)) :
    (dualNumberEquiv.symm d).imI.fst = d.fst.imI 
Arguments: {R : Type u_1} [CommRing R] (d : DualNumber (Quaternion R))
Goal: TrivSqZeroExt.fst (Quaternion.dualNumberEquiv.symm d).imI = (TrivSqZeroExt.fst d).imI
-- Quaternion.fst_imJ_dualNumberEquiv_symm
@[simp]
theorem fst_imJ_dualNumberEquiv_symm (d : DualNumber (Quaternion R)) :
    (dualNumberEquiv.symm d).imJ.fst = d.fst.imJ 
Arguments: {R : Type u_1} [CommRing R] (d : DualNumber (Quaternion R))
Goal: TrivSqZeroExt.fst (Quaternion.dualNumberEquiv.symm d).imJ = (TrivSqZeroExt.fst d).imJ
-- Quaternion.fst_imK_dualNumberEquiv_symm
@[simp]
theorem fst_imK_dualNumberEquiv_symm (d : DualNumber (Quaternion R)) :
    (dualNumberEquiv.symm d).imK.fst = d.fst.imK 
Arguments: {R : Type u_1} [CommRing R] (d : DualNumber (Quaternion R))
Goal: TrivSqZeroExt.fst (Quaternion.dualNumberEquiv.symm d).imK = (TrivSqZeroExt.fst d).imK
-- Quaternion.snd_re_dualNumberEquiv_symm
@[simp]
theorem snd_re_dualNumberEquiv_symm (d : DualNumber (Quaternion R)) :
    (dualNumberEquiv.symm d).re.snd = d.snd.re 
Arguments: {R : Type u_1} [CommRing R] (d : DualNumber (Quaternion R))
Goal: TrivSqZeroExt.snd (Quaternion.dualNumberEquiv.symm d).re = (TrivSqZeroExt.snd d).re
-- Quaternion.snd_imI_dualNumberEquiv_symm
@[simp]
theorem snd_imI_dualNumberEquiv_symm (d : DualNumber (Quaternion R)) :
    (dualNumberEquiv.symm d).imI.snd = d.snd.imI 
Arguments: {R : Type u_1} [CommRing R] (d : DualNumber (Quaternion R))
Goal: TrivSqZeroExt.snd (Quaternion.dualNumberEquiv.symm d).imI = (TrivSqZeroExt.snd d).imI
-- Quaternion.snd_imJ_dualNumberEquiv_symm
@[simp]
theorem snd_imJ_dualNumberEquiv_symm (d : DualNumber (Quaternion R)) :
    (dualNumberEquiv.symm d).imJ.snd = d.snd.imJ 
Arguments: {R : Type u_1} [CommRing R] (d : DualNumber (Quaternion R))
Goal: TrivSqZeroExt.snd (Quaternion.dualNumberEquiv.symm d).imJ = (TrivSqZeroExt.snd d).imJ
-- Quaternion.snd_imK_dualNumberEquiv_symm
@[simp]
theorem snd_imK_dualNumberEquiv_symm (d : DualNumber (Quaternion R)) :
    (dualNumberEquiv.symm d).imK.snd = d.snd.imK 
Arguments: {R : Type u_1} [CommRing R] (d : DualNumber (Quaternion R))
Goal: TrivSqZeroExt.snd (Quaternion.dualNumberEquiv.symm d).imK = (TrivSqZeroExt.snd d).imK
-- ModuleCat.free_map
@[simps]
def free : Type u ⥤ ModuleCat R 
Arguments: (R : Type u) [Ring R] {X Y : Type u} (f : X ⟶ Y)
Goal: (ModuleCat.free R).map f = Finsupp.lmapDomain R R f
-- ModuleCat.free_obj
@[simps]
def free : Type u ⥤ ModuleCat R 
Arguments: (R : Type u) [Ring R] (X : Type u)
Goal: (ModuleCat.free R).obj X = ModuleCat.of R (X →₀ R)
-- ModuleCat.Free.ε_apply
@[simp, nolint simpNF]
theorem ε_apply (r : R) : ε R r = Finsupp.single PUnit.unit r 
Arguments: (R : Type u) [CommRing R] (r : R)
Goal: (ModuleCat.Free.ε R) r = Finsupp.single PUnit.unit r
-- ModuleCat.Free.μ_natural
theorem μ_natural {X Y X' Y' : Type u} (f : X ⟶ Y) (g : X' ⟶ Y') :
    ((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom = (μ R X X').hom ≫ (free R).map (f ⊗ g) 
Arguments: (R : Type u) [CommRing R] {X Y X' Y' : Type u} (f : X ⟶ Y) (g : X' ⟶ Y')
Goal: CategoryTheory.MonoidalCategory.tensorHom ((ModuleCat.free R).map f) ((ModuleCat.free R).map g) ≫
    (ModuleCat.Free.μ R Y Y').hom =
  (ModuleCat.Free.μ R X X').hom ≫ (ModuleCat.free R).map (CategoryTheory.MonoidalCategory.tensorHom f g)
-- ModuleCat.Free.left_unitality
theorem left_unitality (X : Type u) :
    (λ_ ((free R).obj X)).hom =
      (ε R ⊗ 𝟙 ((free R).obj X)) ≫ (μ R (𝟙_ (Type u)) X).hom ≫ map (free R).obj (λ_ X).hom 
Arguments: (R : Type u) [CommRing R] (X : Type u)
Goal: (CategoryTheory.MonoidalCategory.leftUnitor ((ModuleCat.free R).obj X)).hom =
  CategoryTheory.MonoidalCategory.tensorHom (ModuleCat.Free.ε R) (𝟙 ((ModuleCat.free R).obj X)) ≫
    (ModuleCat.Free.μ R (𝟙_ (Type u)) X).hom ≫
      CategoryTheory.map (ModuleCat.free R).obj (CategoryTheory.MonoidalCategory.leftUnitor X).hom
-- ModuleCat.Free.right_unitality
theorem right_unitality (X : Type u) :
    (ρ_ ((free R).obj X)).hom =
      (𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom 
Arguments: (R : Type u) [CommRing R] (X : Type u)
Goal: (CategoryTheory.MonoidalCategory.rightUnitor ((ModuleCat.free R).obj X)).hom =
  CategoryTheory.MonoidalCategory.tensorHom (𝟙 ((ModuleCat.free R).obj X)) (ModuleCat.Free.ε R) ≫
    (ModuleCat.Free.μ R X (𝟙_ (Type u))).hom ≫
      CategoryTheory.map (ModuleCat.free R).obj (CategoryTheory.MonoidalCategory.rightUnitor X).hom
-- ModuleCat.Free.associativity
theorem associativity (X Y Z : Type u) :
    ((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).obj (α_ X Y Z).hom =
      (α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
        (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom 
Arguments: (R : Type u) [CommRing R] (X Y Z : Type u)
Goal: CategoryTheory.MonoidalCategory.tensorHom (ModuleCat.Free.μ R X Y).hom (𝟙 ((ModuleCat.free R).obj Z)) ≫
    (ModuleCat.Free.μ R (CategoryTheory.MonoidalCategory.tensorObj X Y) Z).hom ≫
      CategoryTheory.map (ModuleCat.free R).obj (CategoryTheory.MonoidalCategory.associator X Y Z).hom =
  (CategoryTheory.MonoidalCategory.associator ((ModuleCat.free R).obj X) ((ModuleCat.free R).obj Y)
        ((ModuleCat.free R).obj Z)).hom ≫
    CategoryTheory.MonoidalCategory.tensorHom (𝟙 ((ModuleCat.free R).obj X)) (ModuleCat.Free.μ R Y Z).hom ≫
      (ModuleCat.Free.μ R X (CategoryTheory.MonoidalCategory.tensorObj Y Z)).hom
-- ModuleCat.Free.instLaxMonoidalObjFree_ε
@[simps]
instance : LaxMonoidal.{u} (free R).obj 
Arguments: (R : Type u) [CommRing R]
Goal: CategoryTheory.LaxMonoidal.ε = ModuleCat.Free.ε R
-- ModuleCat.Free.instLaxMonoidalObjFree_μ
@[simps]
instance : LaxMonoidal.{u} (free R).obj 
Arguments: (R : Type u) [CommRing R] (X Y : Type u)
Goal: CategoryTheory.LaxMonoidal.μ X Y = (ModuleCat.Free.μ R X Y).hom
-- CategoryTheory.Free.single_comp_single
theorem single_comp_single {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) (r s : R) :
    (single f r ≫ single g s : Free.of R X ⟶ Free.of R Z) = single (f ≫ g) (r * s) 
Arguments: (R : Type u_1) [CommRing R] (C : Type u) [CategoryTheory.Category.{v, u} C] {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) (r s : R)
Goal: Finsupp.single f r ≫ Finsupp.single g s = Finsupp.single (f ≫ g) (r * s)
-- CategoryTheory.Free.embedding_map
@[simps]
def embedding : C ⥤ Free R C 
Arguments: (R : Type u_1) [CommRing R] (C : Type u) [CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y)
Goal: (CategoryTheory.Free.embedding R C).map f = Finsupp.single f 1
-- CategoryTheory.Free.embedding_obj
@[simps]
def embedding : C ⥤ Free R C 
Arguments: (R : Type u_1) [CommRing R] (C : Type u) [CategoryTheory.Category.{v, u} C] (X : C)
Goal: (CategoryTheory.Free.embedding R C).obj X = X
-- CategoryTheory.Free.lift_obj
@[simps]
def lift (F : C ⥤ D) : Free R C ⥤ D 
Arguments: (R : Type u_1) [CommRing R] {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u} [CategoryTheory.Category.{v, u} D] [CategoryTheory.Preadditive D] [CategoryTheory.Linear R D] (F : C ⥤ D) (X : CategoryTheory.Free R C)
Goal: (CategoryTheory.Free.lift R F).obj X = F.obj X
-- CategoryTheory.Free.lift_map
@[simps]
def lift (F : C ⥤ D) : Free R C ⥤ D 
Arguments: (R : Type u_1) [CommRing R] {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u} [CategoryTheory.Category.{v, u} D] [CategoryTheory.Preadditive D] [CategoryTheory.Linear R D] (F : C ⥤ D) {X Y : CategoryTheory.Free R C} (f : X ⟶ Y)
Goal: (CategoryTheory.Free.lift R F).map f = Finsupp.sum f fun f' r => r • F.map f'
-- CategoryTheory.Free.lift_map_single
theorem lift_map_single (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) (r : R) :
    (lift R F).map (single f r) = r • F.map f 
Arguments: (R : Type u_1) [CommRing R] {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u} [CategoryTheory.Category.{v, u} D] [CategoryTheory.Preadditive D] [CategoryTheory.Linear R D] (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) (r : R)
Goal: (CategoryTheory.Free.lift R F).map (Finsupp.single f r) = r • F.map f
-- HomologicalComplex.i_cyclesMk
@[simp]
lemma i_cyclesMk {i : ι} (x : (forget₂ C Ab).obj (K.X i)) (j : ι) (hj : c.next i = j)
    (hx : ((forget₂ C Ab).map (K.d i j)) x = 0) :
    ((forget₂ C Ab).map (K.iCycles i)) (K.cyclesMk x j hj hx) = x 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.ConcreteCategory C] [CategoryTheory.HasForget₂ C Ab] [CategoryTheory.Abelian C] [(CategoryTheory.forget₂ C Ab).Additive] [(CategoryTheory.forget₂ C Ab).PreservesHomology] {ι : Type u_1} {c : ComplexShape ι} (K : HomologicalComplex C c) {i : ι} (x : ↑((CategoryTheory.forget₂ C Ab).obj (K.X i))) (j : ι) (hj : c.next i = j) (hx : ((CategoryTheory.forget₂ C Ab).map (K.d i j)) x = 0)
Goal: ((CategoryTheory.forget₂ C Ab).map (K.iCycles i)) (K.cyclesMk x j hj hx) = x
-- CategoryTheory.ShortComplex.ShortExact.δ_apply'
lemma δ_apply' (x₃ : (forget₂ C Ab).obj (S.X₃.homology i))
    (x₂ : (forget₂ C Ab).obj (S.X₂.opcycles i))
    (x₁ : (forget₂ C Ab).obj (S.X₁.cycles j))
    (h₂ : (forget₂ C Ab).map (HomologicalComplex.opcyclesMap S.g i) x₂ =
      (forget₂ C Ab).map (S.X₃.homologyι i) x₃)
    (h₁ : (forget₂ C Ab).map (HomologicalComplex.cyclesMap S.f j) x₁ =
      (forget₂ C Ab).map (S.X₂.opcyclesToCycles i j) x₂) :
    (forget₂ C Ab).map (hS.δ i j hij) x₃ = (forget₂ C Ab).map (S.X₁.homologyπ j) x₁ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.ConcreteCategory C] [CategoryTheory.HasForget₂ C Ab] [CategoryTheory.Abelian C] [(CategoryTheory.forget₂ C Ab).Additive] [(CategoryTheory.forget₂ C Ab).PreservesHomology] {ι : Type u_1} {c : ComplexShape ι} {S : CategoryTheory.ShortComplex (HomologicalComplex C c)} (hS : S.ShortExact) (i j : ι) (hij : c.Rel i j) (x₃ : ↑((CategoryTheory.forget₂ C Ab).obj (S.X₃.homology i))) (x₂ : ↑((CategoryTheory.forget₂ C Ab).obj (S.X₂.opcycles i))) (x₁ : ↑((CategoryTheory.forget₂ C Ab).obj (S.X₁.cycles j))) (h₂ :
  ((CategoryTheory.forget₂ C Ab).map (HomologicalComplex.opcyclesMap S.g i)) x₂ =
    ((CategoryTheory.forget₂ C Ab).map (S.X₃.homologyι i)) x₃) (h₁ :
  ((CategoryTheory.forget₂ C Ab).map (HomologicalComplex.cyclesMap S.f j)) x₁ =
    ((CategoryTheory.forget₂ C Ab).map (S.X₂.opcyclesToCycles i j)) x₂)
Goal: ((CategoryTheory.forget₂ C Ab).map (hS.δ i j hij)) x₃ = ((CategoryTheory.forget₂ C Ab).map (S.X₁.homologyπ j)) x₁
-- CategoryTheory.ShortComplex.ShortExact.δ_apply
lemma δ_apply (x₃ : (forget₂ C Ab).obj (S.X₃.X i))
    (hx₃ : (forget₂ C Ab).map (S.X₃.d i j) x₃ = 0)
    (x₂ : (forget₂ C Ab).obj (S.X₂.X i)) (hx₂ : (forget₂ C Ab).map (S.g.f i) x₂ = x₃)
    (x₁ : (forget₂ C Ab).obj (S.X₁.X j))
    (hx₁ : (forget₂ C Ab).map (S.f.f j) x₁ = (forget₂ C Ab).map (S.X₂.d i j) x₂)
    (k : ι) (hk : c.next j = k) :
    (forget₂ C Ab).map (hS.δ i j hij)
      ((forget₂ C Ab).map (S.X₃.homologyπ i) (S.X₃.cyclesMk x₃ j (c.next_eq' hij) hx₃)) =
        (forget₂ C Ab).map (S.X₁.homologyπ j) (S.X₁.cyclesMk x₁ k hk (by
          have := hS.mono_f
          apply (Preadditive.mono_iff_injective (S.f.f k)).1 inferInstance
          erw [← forget₂_comp_apply, ← HomologicalComplex.Hom.comm, forget₂_comp_apply, hx₁,
            ← forget₂_comp_apply, HomologicalComplex.d_comp_d, Functor.map_zero, map_zero,
            AddMonoidHom.zero_apply])) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.ConcreteCategory C] [CategoryTheory.HasForget₂ C Ab] [CategoryTheory.Abelian C] [(CategoryTheory.forget₂ C Ab).Additive] [(CategoryTheory.forget₂ C Ab).PreservesHomology] {ι : Type u_1} {c : ComplexShape ι} {S : CategoryTheory.ShortComplex (HomologicalComplex C c)} (hS : S.ShortExact) (i j : ι) (hij : c.Rel i j) (x₃ : ↑((CategoryTheory.forget₂ C Ab).obj (S.X₃.X i))) (hx₃ : ((CategoryTheory.forget₂ C Ab).map (S.X₃.d i j)) x₃ = 0) (x₂ : ↑((CategoryTheory.forget₂ C Ab).obj (S.X₂.X i))) (hx₂ : ((CategoryTheory.forget₂ C Ab).map (S.g.f i)) x₂ = x₃) (x₁ : ↑((CategoryTheory.forget₂ C Ab).obj (S.X₁.X j))) (hx₁ : ((CategoryTheory.forget₂ C Ab).map (S.f.f j)) x₁ = ((CategoryTheory.forget₂ C Ab).map (S.X₂.d i j)) x₂) (k : ι) (hk : c.next j = k)
Goal: ((CategoryTheory.forget₂ C Ab).map (hS.δ i j hij))
    (((CategoryTheory.forget₂ C Ab).map (S.X₃.homologyπ i)) (S.X₃.cyclesMk x₃ j ⋯ hx₃)) =
  ((CategoryTheory.forget₂ C Ab).map (S.X₁.homologyπ j)) (S.X₁.cyclesMk x₁ k hk ⋯)
-- AlgebraCat.hasLimitsOfSize
lemma hasLimitsOfSize [UnivLE.{v, w}] : HasLimitsOfSize.{t, v} (AlgebraCat.{w} R) 
Arguments: {R : Type u} [CommRing R] [UnivLE.{v, w}]
Goal: CategoryTheory.Limits.HasLimitsOfSize.{t, v, w, max (w + 1) u} (AlgebraCat R)
-- HomologicalComplex.single_obj_X_self
@[simp]
lemma single_obj_X_self (j : ι) (A : V) :
    ((single V c j).obj A).X j = A 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) (A : V)
Goal: ((HomologicalComplex.single V c j).obj A).X j = A
-- HomologicalComplex.isZero_single_obj_X
lemma isZero_single_obj_X (j : ι) (A : V) (i : ι) (hi : i ≠ j) :
    IsZero (((single V c j).obj A).X i) 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) (A : V) (i : ι) (hi : i ≠ j)
Goal: CategoryTheory.Limits.IsZero (((HomologicalComplex.single V c j).obj A).X i)
-- HomologicalComplex.single_obj_d
@[simp]
lemma single_obj_d (j : ι) (A : V) (k l : ι) :
    ((single V c j).obj A).d k l = 0 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) (A : V) (k l : ι)
Goal: ((HomologicalComplex.single V c j).obj A).d k l = 0
-- HomologicalComplex.single_map_f_self
@[reassoc]
theorem single_map_f_self (j : ι) {A B : V} (f : A ⟶ B) :
    ((single V c j).map f).f j = (singleObjXSelf c j A).hom ≫
      f ≫ (singleObjXSelf c j B).inv 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) {A B : V} (f : A ⟶ B)
Goal: ((HomologicalComplex.single V c j).map f).f j =
  (HomologicalComplex.singleObjXSelf c j A).hom ≫ f ≫ (HomologicalComplex.singleObjXSelf c j B).inv
-- HomologicalComplex.singleCompEvalIsoSelf_inv_app
@[simps!]
noncomputable def singleCompEvalIsoSelf (j : ι) : single V c j ⋙ eval V c j ≅ 𝟭 V 
Arguments: (V : Type u) [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) (X : V)
Goal: (HomologicalComplex.singleCompEvalIsoSelf V c j).inv.app X = (HomologicalComplex.singleObjXSelf c j X).inv
-- HomologicalComplex.singleCompEvalIsoSelf_hom_app
@[simps!]
noncomputable def singleCompEvalIsoSelf (j : ι) : single V c j ⋙ eval V c j ≅ 𝟭 V 
Arguments: (V : Type u) [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) (X : V)
Goal: (HomologicalComplex.singleCompEvalIsoSelf V c j).hom.app X = (HomologicalComplex.singleObjXSelf c j X).hom
-- HomologicalComplex.isZero_single_comp_eval
lemma isZero_single_comp_eval (j i : ι) (hi : i ≠ j) : IsZero (single V c j ⋙ eval V c i) 
Arguments: (V : Type u) [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j i : ι) (hi : i ≠ j)
Goal: CategoryTheory.Limits.IsZero (HomologicalComplex.single V c j ⋙ HomologicalComplex.eval V c i)
-- HomologicalComplex.from_single_hom_ext
@[ext]
lemma from_single_hom_ext {K : HomologicalComplex V c} {j : ι} {A : V}
    {f g : (single V c j).obj A ⟶ K} (hfg : f.f j = g.f j) : f = g 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {ι : Type u_1} [DecidableEq ι] {c : ComplexShape ι} {K : HomologicalComplex V c} {j : ι} {A : V} {f g : (HomologicalComplex.single V c j).obj A ⟶ K} (hfg : f.f j = g.f j)
Goal: f = g
-- HomologicalComplex.to_single_hom_ext
@[ext]
lemma to_single_hom_ext {K : HomologicalComplex V c} {j : ι} {A : V}
    {f g : K ⟶ (single V c j).obj A} (hfg : f.f j = g.f j) : f = g 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {ι : Type u_1} [DecidableEq ι] {c : ComplexShape ι} {K : HomologicalComplex V c} {j : ι} {A : V} {f g : K ⟶ (HomologicalComplex.single V c j).obj A} (hfg : f.f j = g.f j)
Goal: f = g
-- HomologicalComplex.mkHomToSingle_f
@[simp]
lemma mkHomToSingle_f {K : HomologicalComplex V c} {j : ι} {A : V} (φ : K.X j ⟶ A)
    (hφ : ∀ (i : ι), c.Rel i j → K.d i j ≫ φ = 0) :
    (mkHomToSingle φ hφ).f j = φ ≫ (singleObjXSelf c j A).inv 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {ι : Type u_1} [DecidableEq ι] {c : ComplexShape ι} {K : HomologicalComplex V c} {j : ι} {A : V} (φ : K.X j ⟶ A) (hφ : ∀ (i : ι), c.Rel i j → K.d i j ≫ φ = 0)
Goal: (HomologicalComplex.mkHomToSingle φ hφ).f j = φ ≫ (HomologicalComplex.singleObjXSelf c j A).inv
-- HomologicalComplex.mkHomFromSingle_f
@[simp]
lemma mkHomFromSingle_f {K : HomologicalComplex V c} {j : ι} {A : V} (φ : A ⟶ K.X j)
    (hφ : ∀ (k : ι), c.Rel j k → φ ≫ K.d j k = 0) :
    (mkHomFromSingle φ hφ).f j = (singleObjXSelf c j A).hom ≫ φ 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {ι : Type u_1} [DecidableEq ι] {c : ComplexShape ι} {K : HomologicalComplex V c} {j : ι} {A : V} (φ : A ⟶ K.X j) (hφ : ∀ (k : ι), c.Rel j k → φ ≫ K.d j k = 0)
Goal: (HomologicalComplex.mkHomFromSingle φ hφ).f j = (HomologicalComplex.singleObjXSelf c j A).hom ≫ φ
-- ChainComplex.single₀_obj_zero
@[simp]
lemma single₀_obj_zero (A : V) :
    ((single₀ V).obj A).X 0 = A 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] (A : V)
Goal: ((ChainComplex.single₀ V).obj A).X 0 = A
-- ChainComplex.single₀_map_f_zero
@[simp]
lemma single₀_map_f_zero {A B : V} (f : A ⟶ B) :
    ((single₀ V).map f).f 0 = f 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] {A B : V} (f : A ⟶ B)
Goal: ((ChainComplex.single₀ V).map f).f 0 = f
-- ChainComplex.single₀ObjXSelf
@[simp]
lemma single₀ObjXSelf (X : V) :
    HomologicalComplex.singleObjXSelf (ComplexShape.down ℕ) 0 X = Iso.refl _ 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] [CategoryTheory.Limits.HasZeroObject V] (X : V)
Goal: HomologicalComplex.singleObjXSelf (ComplexShape.down ℕ) 0 X =
  CategoryTheory.Iso.refl (((HomologicalComplex.single V (ComplexShape.down ℕ) 0).obj X).X 0)