ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- DivisionRing.ratCast_def
class DivisionRing (α : Type*)
  extends Ring α, DivInvMonoid α, Nontrivial α, NNRatCast α, RatCast α where
    protected mul_inv_cancel : ∀ (a : α), a ≠ 0 → a * a⁻¹ = 1
    protected inv_zero : (0 : α)⁻¹ = 0
  protected nnratCast := NNRat.castRec
    protected nnratCast_def (q : ℚ≥0) : (NNRat.cast q : α) = q.num / q.den := by intros; rfl
    protected nnqsmul : ℚ≥0 → α → α
    protected nnqsmul_def (q : ℚ≥0) (a : α) : nnqsmul q a = NNRat.cast q * a := by intros; rfl
  protected ratCast := Rat.castRec
    protected ratCast_def (q : ℚ) : (Rat.cast q : α) = q.num / q.den := by intros; rfl
    protected qsmul : ℚ → α → α
    protected qsmul_def (a : ℚ) (x : α) : qsmul a x = Rat.cast a * x := by intros; rfl
Arguments: {α : Type u_4} [self : DivisionRing α] (q : ℚ)
Goal: ↑q = ↑q.num / ↑q.den
-- DivisionRing.nnqsmul_def
class DivisionRing (α : Type*)
  extends Ring α, DivInvMonoid α, Nontrivial α, NNRatCast α, RatCast α where
    protected mul_inv_cancel : ∀ (a : α), a ≠ 0 → a * a⁻¹ = 1
    protected inv_zero : (0 : α)⁻¹ = 0
  protected nnratCast := NNRat.castRec
    protected nnratCast_def (q : ℚ≥0) : (NNRat.cast q : α) = q.num / q.den := by intros; rfl
    protected nnqsmul : ℚ≥0 → α → α
    protected nnqsmul_def (q : ℚ≥0) (a : α) : nnqsmul q a = NNRat.cast q * a := by intros; rfl
  protected ratCast := Rat.castRec
    protected ratCast_def (q : ℚ) : (Rat.cast q : α) = q.num / q.den := by intros; rfl
    protected qsmul : ℚ → α → α
    protected qsmul_def (a : ℚ) (x : α) : qsmul a x = Rat.cast a * x := by intros; rfl
Arguments: {α : Type u_4} [self : DivisionRing α] (q : ℚ≥0) (a : α)
Goal: DivisionRing.nnqsmul q a = ↑q * a
-- DivisionRing.nnratCast_def
class DivisionRing (α : Type*)
  extends Ring α, DivInvMonoid α, Nontrivial α, NNRatCast α, RatCast α where
    protected mul_inv_cancel : ∀ (a : α), a ≠ 0 → a * a⁻¹ = 1
    protected inv_zero : (0 : α)⁻¹ = 0
  protected nnratCast := NNRat.castRec
    protected nnratCast_def (q : ℚ≥0) : (NNRat.cast q : α) = q.num / q.den := by intros; rfl
    protected nnqsmul : ℚ≥0 → α → α
    protected nnqsmul_def (q : ℚ≥0) (a : α) : nnqsmul q a = NNRat.cast q * a := by intros; rfl
  protected ratCast := Rat.castRec
    protected ratCast_def (q : ℚ) : (Rat.cast q : α) = q.num / q.den := by intros; rfl
    protected qsmul : ℚ → α → α
    protected qsmul_def (a : ℚ) (x : α) : qsmul a x = Rat.cast a * x := by intros; rfl
Arguments: {α : Type u_4} [self : DivisionRing α] (q : ℚ≥0)
Goal: ↑q = ↑q.num / ↑q.den
-- DivisionRing.mul_inv_cancel
class DivisionRing (α : Type*)
  extends Ring α, DivInvMonoid α, Nontrivial α, NNRatCast α, RatCast α where
    protected mul_inv_cancel : ∀ (a : α), a ≠ 0 → a * a⁻¹ = 1
    protected inv_zero : (0 : α)⁻¹ = 0
  protected nnratCast := NNRat.castRec
    protected nnratCast_def (q : ℚ≥0) : (NNRat.cast q : α) = q.num / q.den := by intros; rfl
    protected nnqsmul : ℚ≥0 → α → α
    protected nnqsmul_def (q : ℚ≥0) (a : α) : nnqsmul q a = NNRat.cast q * a := by intros; rfl
  protected ratCast := Rat.castRec
    protected ratCast_def (q : ℚ) : (Rat.cast q : α) = q.num / q.den := by intros; rfl
    protected qsmul : ℚ → α → α
    protected qsmul_def (a : ℚ) (x : α) : qsmul a x = Rat.cast a * x := by intros; rfl
Arguments: {α : Type u_4} [self : DivisionRing α] (a : α)
Goal: a ≠ 0 → a * a⁻¹ = 1
-- NNRat.cast_def
lemma cast_def (q : ℚ≥0) : (q : α) = q.num / q.den 
Arguments: {α : Type u_1} [DivisionSemiring α] (q : ℚ≥0)
Goal: ↑q = ↑q.num / ↑q.den
-- NNRat.smul_def
lemma smul_def (q : ℚ≥0) (a : α) : q • a = q * a 
Arguments: {α : Type u_1} [DivisionSemiring α] (q : ℚ≥0) (a : α)
Goal: q • a = ↑q * a
-- NNRat.smul_one_eq_cast
@[simp] lemma smul_one_eq_cast (q : ℚ≥0) : q • (1 : α) = q 
Arguments: (α : Type u_1) [DivisionSemiring α] (q : ℚ≥0)
Goal: q • 1 = ↑q
-- Rat.cast_def
lemma cast_def (q : ℚ) : (q : K) = q.num / q.den 
Arguments: {K : Type u_3} [DivisionRing K] (q : ℚ)
Goal: ↑q = ↑q.num / ↑q.den
-- Rat.cast_mk'
lemma cast_mk' (a b h1 h2) : ((⟨a, b, h1, h2⟩ : ℚ) : K) = a / b 
Arguments: {K : Type u_3} [DivisionRing K] (a : ℤ) (b : ℕ) (h1 : b ≠ 0) (h2 : a.natAbs.Coprime b)
Goal: ↑{ num := a, den := b, den_nz := h1, reduced := h2 } = ↑a / ↑b
-- Rat.smul_def
theorem smul_def (a : ℚ) (x : K) : a • x = ↑a * x 
Arguments: {K : Type u_3} [DivisionRing K] (a : ℚ) (x : K)
Goal: a • x = ↑a * x
-- Rat.smul_one_eq_cast
@[simp]
theorem smul_one_eq_cast (A : Type*) [DivisionRing A] (m : ℚ) : m • (1 : A) = ↑m 
Arguments: (A : Type u_4) [DivisionRing A] (m : ℚ)
Goal: m • 1 = ↑m
-- Rat.ofScientific_eq_ofScientific
@[simp]
theorem Rat.ofScientific_eq_ofScientific (m : ℕ) (s : Bool) (e : ℕ) :
    Rat.ofScientific (OfNat.ofNat m) s (OfNat.ofNat e) = OfScientific.ofScientific m s e 
Arguments: (m : ℕ) (s : Bool) (e : ℕ)
Goal: Rat.ofScientific (OfNat.ofNat m) s (OfNat.ofNat e) = OfScientific.ofScientific m s e
-- Odd.neg_zpow
theorem Odd.neg_zpow (h : Odd n) (a : α) : (-a) ^ n = -a ^ n 
Arguments: {α : Type u_1} [DivisionRing α] {n : ℤ} (h : Odd n) (a : α)
Goal: (-a) ^ n = -a ^ n
-- Odd.neg_one_zpow
theorem Odd.neg_one_zpow (h : Odd n) : (-1 : α) ^ n = -1 
Arguments: {α : Type u_1} [DivisionRing α] {n : ℤ} (h : Odd n)
Goal: (-1) ^ n = -1
-- AlgebraCat.instMonoidalCategory.tensorObj_carrier
@[simps!]
noncomputable abbrev tensorObj (X Y : AlgebraCat.{u} R) : AlgebraCat.{u} R 
Arguments: {R : Type u} [CommRing R] (X Y : AlgebraCat R)
Goal: ↑(AlgebraCat.instMonoidalCategory.tensorObj X Y) = TensorProduct R ↑X ↑Y
-- AlgebraCat.forget₂_map_associator_hom
theorem forget₂_map_associator_hom (X Y Z : AlgebraCat.{u} R) :
    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =
      (α_
        (forget₂ _ (ModuleCat R) |>.obj X)
        (forget₂ _ (ModuleCat R) |>.obj Y)
        (forget₂ _ (ModuleCat R) |>.obj Z)).hom 
Arguments: {R : Type u} [CommRing R] (X Y Z : AlgebraCat R)
Goal: (CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =
  (α_ ((CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R)).obj X)
      ((CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R)).obj Y)
      ((CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R)).obj Z)).hom
-- AlgebraCat.forget₂_map_associator_inv
theorem forget₂_map_associator_inv (X Y Z : AlgebraCat.{u} R) :
    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).inv =
      (α_
        (forget₂ _ (ModuleCat R) |>.obj X)
        (forget₂ _ (ModuleCat R) |>.obj Y)
        (forget₂ _ (ModuleCat R) |>.obj Z)).inv 
Arguments: {R : Type u} [CommRing R] (X Y Z : AlgebraCat R)
Goal: (CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).inv =
  (α_ ((CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R)).obj X)
      ((CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R)).obj Y)
      ((CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R)).obj Z)).inv
-- HomologicalComplex₂.totalFlipIsoX_hom_D₁
@[reassoc]
lemma totalFlipIsoX_hom_D₁ (j j' : J) :
    (K.totalFlipIsoX c j).hom ≫ K.D₁ c j j' =
      K.flip.D₂ c j j' ≫ (K.totalFlipIsoX c j').hom 
Arguments: {C : Type u_1} {I₁ : Type u_2} {I₂ : Type u_3} {J : Type u_4} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c : ComplexShape J) [TotalComplexShape c₁ c₂ c] [TotalComplexShape c₂ c₁ c] [TotalComplexShapeSymmetry c₁ c₂ c] [K.HasTotal c] [K.flip.HasTotal c] [DecidableEq J] (j j' : J)
Goal: (K.totalFlipIsoX c j).hom ≫ K.D₁ c j j' = K.flip.D₂ c j j' ≫ (K.totalFlipIsoX c j').hom
-- HomologicalComplex₂.totalFlipIsoX_hom_D₂
@[reassoc]
lemma totalFlipIsoX_hom_D₂ (j j' : J) :
    (K.totalFlipIsoX c j).hom ≫ K.D₂ c j j' =
      K.flip.D₁ c j j' ≫ (K.totalFlipIsoX c j').hom 
Arguments: {C : Type u_1} {I₁ : Type u_2} {I₂ : Type u_3} {J : Type u_4} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c : ComplexShape J) [TotalComplexShape c₁ c₂ c] [TotalComplexShape c₂ c₁ c] [TotalComplexShapeSymmetry c₁ c₂ c] [K.HasTotal c] [K.flip.HasTotal c] [DecidableEq J] (j j' : J)
Goal: (K.totalFlipIsoX c j).hom ≫ K.D₂ c j j' = K.flip.D₁ c j j' ≫ (K.totalFlipIsoX c j').hom
-- HomologicalComplex₂.totalFlipIso_hom_f_D₁
@[reassoc]
lemma totalFlipIso_hom_f_D₁ (j j' : J) :
    (K.totalFlipIso c).hom.f j ≫ K.D₁ c j j' =
      K.flip.D₂ c j j' ≫ (K.totalFlipIso c).hom.f j' 
Arguments: {C : Type u_1} {I₁ : Type u_2} {I₂ : Type u_3} {J : Type u_4} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c : ComplexShape J) [TotalComplexShape c₁ c₂ c] [TotalComplexShape c₂ c₁ c] [TotalComplexShapeSymmetry c₁ c₂ c] [K.HasTotal c] [K.flip.HasTotal c] [DecidableEq J] (j j' : J)
Goal: (K.totalFlipIso c).hom.f j ≫ K.D₁ c j j' = K.flip.D₂ c j j' ≫ (K.totalFlipIso c).hom.f j'
-- HomologicalComplex₂.totalFlipIso_hom_f_D₂
@[reassoc]
lemma totalFlipIso_hom_f_D₂ (j j' : J) :
    (K.totalFlipIso c).hom.f j ≫ K.D₂ c j j' =
      K.flip.D₁ c j j' ≫ (K.totalFlipIso c).hom.f j' 
Arguments: {C : Type u_1} {I₁ : Type u_2} {I₂ : Type u_3} {J : Type u_4} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c : ComplexShape J) [TotalComplexShape c₁ c₂ c] [TotalComplexShape c₂ c₁ c] [TotalComplexShapeSymmetry c₁ c₂ c] [K.HasTotal c] [K.flip.HasTotal c] [DecidableEq J] (j j' : J)
Goal: (K.totalFlipIso c).hom.f j ≫ K.D₂ c j j' = K.flip.D₁ c j j' ≫ (K.totalFlipIso c).hom.f j'
-- HomologicalComplex₂.ιTotal_totalFlipIso_f_hom
@[reassoc (attr := simp)]
lemma ιTotal_totalFlipIso_f_hom
    (i₁ : I₁) (i₂ : I₂) (j : J) (h : ComplexShape.π c₂ c₁ c (i₂, i₁) = j) :
    K.flip.ιTotal c i₂ i₁ j h ≫ (K.totalFlipIso c).hom.f j =
      ComplexShape.σ c₁ c₂ c i₁ i₂ • K.ιTotal c i₁ i₂ j
        (by rw [← ComplexShape.π_symm c₁ c₂ c i₁ i₂, h]) 
Arguments: {C : Type u_1} {I₁ : Type u_2} {I₂ : Type u_3} {J : Type u_4} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c : ComplexShape J) [TotalComplexShape c₁ c₂ c] [TotalComplexShape c₂ c₁ c] [TotalComplexShapeSymmetry c₁ c₂ c] [K.HasTotal c] [K.flip.HasTotal c] [DecidableEq J] (i₁ : I₁) (i₂ : I₂) (j : J) (h : c₂.π c₁ c (i₂, i₁) = j)
Goal: K.flip.ιTotal c i₂ i₁ j h ≫ (K.totalFlipIso c).hom.f j = c₁.σ c₂ c i₁ i₂ • K.ιTotal c i₁ i₂ j ⋯
-- HomologicalComplex₂.ιTotal_totalFlipIso_f_inv
@[reassoc (attr := simp)]
lemma ιTotal_totalFlipIso_f_inv
    (i₁ : I₁) (i₂ : I₂) (j : J) (h : ComplexShape.π c₁ c₂ c (i₁, i₂) = j) :
    K.ιTotal c i₁ i₂ j h ≫ (K.totalFlipIso c).inv.f j =
      ComplexShape.σ c₁ c₂ c i₁ i₂ • K.flip.ιTotal c i₂ i₁ j
        (by rw [ComplexShape.π_symm c₁ c₂ c i₁ i₂, h]) 
Arguments: {C : Type u_1} {I₁ : Type u_2} {I₂ : Type u_3} {J : Type u_4} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c : ComplexShape J) [TotalComplexShape c₁ c₂ c] [TotalComplexShape c₂ c₁ c] [TotalComplexShapeSymmetry c₁ c₂ c] [K.HasTotal c] [K.flip.HasTotal c] [DecidableEq J] (i₁ : I₁) (i₂ : I₂) (j : J) (h : c₁.π c₂ c (i₁, i₂) = j)
Goal: K.ιTotal c i₁ i₂ j h ≫ (K.totalFlipIso c).inv.f j = c₁.σ c₂ c i₁ i₂ • K.flip.ιTotal c i₂ i₁ j ⋯
-- HomologicalComplex₂.flip_totalFlipIso
lemma flip_totalFlipIso : K.flip.totalFlipIso c = (K.totalFlipIso c).symm 
Arguments: {C : Type u_1} {I₁ : Type u_2} {I₂ : Type u_3} {J : Type u_4} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Preadditive C] {c₁ : ComplexShape I₁} {c₂ : ComplexShape I₂} (K : HomologicalComplex₂ C c₁ c₂) (c : ComplexShape J) [TotalComplexShape c₁ c₂ c] [TotalComplexShape c₂ c₁ c] [TotalComplexShapeSymmetry c₁ c₂ c] [K.HasTotal c] [K.flip.HasTotal c] [DecidableEq J] [TotalComplexShapeSymmetry c₂ c₁ c] [TotalComplexShapeSymmetrySymmetry c₁ c₂ c]
Goal: K.flip.totalFlipIso c = (K.totalFlipIso c).symm
-- ULift.up_nnratCast
@[simp, norm_cast] lemma up_nnratCast [NNRatCast α] (q : ℚ≥0) : up (q : α) = q 
Arguments: {α : Type u} [NNRatCast α] (q : ℚ≥0)
Goal: { down := ↑q } = ↑q
-- ULift.down_nnratCast
@[simp, norm_cast] lemma down_nnratCast [NNRatCast α] (q : ℚ≥0) : down (q : ULift α) = q 
Arguments: {α : Type u} [NNRatCast α] (q : ℚ≥0)
Goal: (↑q).down = ↑q
-- ULift.up_ratCast
@[simp, norm_cast] lemma up_ratCast [RatCast α] (q : ℚ) : up (q : α) = q 
Arguments: {α : Type u} [RatCast α] (q : ℚ)
Goal: { down := ↑q } = ↑q
-- ULift.down_ratCast
@[simp, norm_cast] lemma down_ratCast [RatCast α] (q : ℚ) : down (q : ULift α) = q 
Arguments: {α : Type u} [RatCast α] (q : ℚ)
Goal: (↑q).down = ↑q
-- MulEquiv.isField
protected theorem isField {A : Type*} (B : Type*) [Semiring A] [Semiring B] (hB : IsField B)
    (e : A ≃* B) : IsField A 
Arguments: {A : Type u_1} (B : Type u_2) [Semiring A] [Semiring B] (hB : IsField B) (e : A ≃* B)
Goal: IsField A
-- HomologicalComplex.exactAt_single_obj
lemma exactAt_single_obj (A : C) (i : ι) (hi : i ≠ j) :
    ExactAt ((single C c j).obj A) i 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) (A : C) (i : ι) (hi : i ≠ j)
Goal: ((HomologicalComplex.single C c j).obj A).ExactAt i
-- HomologicalComplex.isZero_single_obj_homology
lemma isZero_single_obj_homology (A : C) (i : ι) (hi : i ≠ j) :
    IsZero (((single C c j).obj A).homology i) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) (A : C) (i : ι) (hi : i ≠ j)
Goal: CategoryTheory.Limits.IsZero (((HomologicalComplex.single C c j).obj A).homology i)
-- HomologicalComplex.singleObjCyclesSelfIso_hom
@[reassoc]
lemma singleObjCyclesSelfIso_hom :
    (singleObjCyclesSelfIso c j A).hom =
      ((single C c j).obj A).iCycles j ≫ (singleObjXSelf c j A).hom 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) (A : C)
Goal: (HomologicalComplex.singleObjCyclesSelfIso c j A).hom =
  ((HomologicalComplex.single C c j).obj A).iCycles j ≫ (HomologicalComplex.singleObjXSelf c j A).hom
-- HomologicalComplex.singleObjOpcyclesSelfIso_hom
@[reassoc]
lemma singleObjOpcyclesSelfIso_hom :
    (singleObjOpcyclesSelfIso c j A).hom =
      (singleObjXSelf c j A).inv ≫ ((single C c j).obj A).pOpcycles j 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) (A : C)
Goal: (HomologicalComplex.singleObjOpcyclesSelfIso c j A).hom =
  (HomologicalComplex.singleObjXSelf c j A).inv ≫ ((HomologicalComplex.single C c j).obj A).pOpcycles j
-- HomologicalComplex.singleObjCyclesSelfIso_inv_iCycles
@[reassoc (attr := simp)]
lemma singleObjCyclesSelfIso_inv_iCycles :
    (singleObjCyclesSelfIso _ _ _).inv ≫ ((single C c j).obj A).iCycles j =
      (singleObjXSelf c j A).inv 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) (A : C)
Goal: (HomologicalComplex.singleObjCyclesSelfIso c j A).inv ≫ ((HomologicalComplex.single C c j).obj A).iCycles j =
  (HomologicalComplex.singleObjXSelf c j A).inv
-- HomologicalComplex.homologyπ_singleObjHomologySelfIso_hom
@[reassoc (attr := simp)]
lemma homologyπ_singleObjHomologySelfIso_hom :
    ((single C c j).obj A).homologyπ j ≫ (singleObjHomologySelfIso _ _ _).hom =
      (singleObjCyclesSelfIso _ _ _).hom 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) (A : C)
Goal: ((HomologicalComplex.single C c j).obj A).homologyπ j ≫ (HomologicalComplex.singleObjHomologySelfIso c j A).hom =
  (HomologicalComplex.singleObjCyclesSelfIso c j A).hom
-- HomologicalComplex.singleObjHomologySelfIso_hom_singleObjHomologySelfIso_inv
@[reassoc (attr := simp)]
lemma singleObjHomologySelfIso_hom_singleObjHomologySelfIso_inv :
    (singleObjCyclesSelfIso c j A).hom ≫ (singleObjHomologySelfIso c j A).inv =
      ((single C c j).obj A).homologyπ j 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) (A : C)
Goal: (HomologicalComplex.singleObjCyclesSelfIso c j A).hom ≫ (HomologicalComplex.singleObjHomologySelfIso c j A).inv =
  ((HomologicalComplex.single C c j).obj A).homologyπ j
-- HomologicalComplex.singleObjCyclesSelfIso_hom_singleObjOpcyclesSelfIso_hom
@[reassoc (attr := simp)]
lemma singleObjCyclesSelfIso_hom_singleObjOpcyclesSelfIso_hom :
    (singleObjCyclesSelfIso c j A).hom ≫ (singleObjOpcyclesSelfIso c j A).hom =
      ((single C c j).obj A).iCycles j ≫ ((single C c j).obj A).pOpcycles j 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) (A : C)
Goal: (HomologicalComplex.singleObjCyclesSelfIso c j A).hom ≫ (HomologicalComplex.singleObjOpcyclesSelfIso c j A).hom =
  ((HomologicalComplex.single C c j).obj A).iCycles j ≫ ((HomologicalComplex.single C c j).obj A).pOpcycles j
-- HomologicalComplex.singleObjCyclesSelfIso_inv_homologyπ
@[reassoc (attr := simp)]
lemma singleObjCyclesSelfIso_inv_homologyπ  :
    (singleObjCyclesSelfIso _ _ _).inv ≫ ((single C c j).obj A).homologyπ j =
      (singleObjHomologySelfIso _ _ _).inv 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) (A : C)
Goal: (HomologicalComplex.singleObjCyclesSelfIso c j A).inv ≫ ((HomologicalComplex.single C c j).obj A).homologyπ j =
  (HomologicalComplex.singleObjHomologySelfIso c j A).inv
-- HomologicalComplex.singleObjHomologySelfIso_inv_homologyι
@[reassoc (attr := simp)]
lemma singleObjHomologySelfIso_inv_homologyι :
    (singleObjHomologySelfIso _ _ _).inv ≫ ((single C c j).obj A).homologyι j =
      (singleObjOpcyclesSelfIso _ _ _).hom 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) (A : C)
Goal: (HomologicalComplex.singleObjHomologySelfIso c j A).inv ≫ ((HomologicalComplex.single C c j).obj A).homologyι j =
  (HomologicalComplex.singleObjOpcyclesSelfIso c j A).hom
-- HomologicalComplex.homologyι_singleObjOpcyclesSelfIso_inv
@[reassoc (attr := simp)]
lemma homologyι_singleObjOpcyclesSelfIso_inv :
    ((single C c j).obj A).homologyι j ≫ (singleObjOpcyclesSelfIso _ _ _).inv =
      (singleObjHomologySelfIso _ _ _).hom 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) (A : C)
Goal: ((HomologicalComplex.single C c j).obj A).homologyι j ≫ (HomologicalComplex.singleObjOpcyclesSelfIso c j A).inv =
  (HomologicalComplex.singleObjHomologySelfIso c j A).hom
-- HomologicalComplex.singleObjHomologySelfIso_hom_singleObjOpcyclesSelfIso_hom
@[reassoc (attr := simp)]
lemma singleObjHomologySelfIso_hom_singleObjOpcyclesSelfIso_hom :
    (singleObjHomologySelfIso _ _ _).hom ≫ (singleObjOpcyclesSelfIso _ _ _).hom =
      ((single C c j).obj A).homologyι j 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) (A : C)
Goal: (HomologicalComplex.singleObjHomologySelfIso c j A).hom ≫ (HomologicalComplex.singleObjOpcyclesSelfIso c j A).hom =
  ((HomologicalComplex.single C c j).obj A).homologyι j
-- HomologicalComplex.singleObjCyclesSelfIso_hom_naturality
@[reassoc (attr := simp)]
lemma singleObjCyclesSelfIso_hom_naturality :
    cyclesMap ((single C c j).map f) j ≫ (singleObjCyclesSelfIso c j B).hom =
      (singleObjCyclesSelfIso c j A).hom ≫ f 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) {A B : C} (f : A ⟶ B)
Goal: HomologicalComplex.cyclesMap ((HomologicalComplex.single C c j).map f) j ≫
    (HomologicalComplex.singleObjCyclesSelfIso c j B).hom =
  (HomologicalComplex.singleObjCyclesSelfIso c j A).hom ≫ f
-- HomologicalComplex.singleObjCyclesSelfIso_inv_naturality
@[reassoc (attr := simp)]
lemma singleObjCyclesSelfIso_inv_naturality :
    (singleObjCyclesSelfIso c j A).inv ≫ cyclesMap ((single C c j).map f) j =
      f ≫ (singleObjCyclesSelfIso c j B).inv 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) {A B : C} (f : A ⟶ B)
Goal: (HomologicalComplex.singleObjCyclesSelfIso c j A).inv ≫
    HomologicalComplex.cyclesMap ((HomologicalComplex.single C c j).map f) j =
  f ≫ (HomologicalComplex.singleObjCyclesSelfIso c j B).inv
-- HomologicalComplex.singleObjHomologySelfIso_hom_naturality
@[reassoc (attr := simp)]
lemma singleObjHomologySelfIso_hom_naturality :
    homologyMap ((single C c j).map f) j ≫ (singleObjHomologySelfIso c j B).hom =
      (singleObjHomologySelfIso c j A).hom ≫ f 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) {A B : C} (f : A ⟶ B)
Goal: HomologicalComplex.homologyMap ((HomologicalComplex.single C c j).map f) j ≫
    (HomologicalComplex.singleObjHomologySelfIso c j B).hom =
  (HomologicalComplex.singleObjHomologySelfIso c j A).hom ≫ f
-- HomologicalComplex.singleObjHomologySelfIso_inv_naturality
@[reassoc (attr := simp)]
lemma singleObjHomologySelfIso_inv_naturality :
    (singleObjHomologySelfIso c j A).inv ≫ homologyMap ((single C c j).map f) j =
      f ≫ (singleObjHomologySelfIso c j B).inv 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) {A B : C} (f : A ⟶ B)
Goal: (HomologicalComplex.singleObjHomologySelfIso c j A).inv ≫
    HomologicalComplex.homologyMap ((HomologicalComplex.single C c j).map f) j =
  f ≫ (HomologicalComplex.singleObjHomologySelfIso c j B).inv
-- HomologicalComplex.singleObjOpcyclesSelfIso_hom_naturality
@[reassoc (attr := simp)]
lemma singleObjOpcyclesSelfIso_hom_naturality :
    (singleObjOpcyclesSelfIso c j A).hom ≫ opcyclesMap ((single C c j).map f) j  =
      f ≫ (singleObjOpcyclesSelfIso c j B).hom 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroObject C] {ι : Type u_1} [DecidableEq ι] (c : ComplexShape ι) (j : ι) {A B : C} (f : A ⟶ B)
Goal: (HomologicalComplex.singleObjOpcyclesSelfIso c j A).hom ≫
    HomologicalComplex.opcyclesMap ((HomologicalComplex.single C c j).map f) j =
  f ≫ (HomologicalComplex.singleObjOpcyclesSelfIso c j B).hom