ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HomologicalComplex.pOpcycles_opcyclesIsoSc'_hom
@[reassoc (attr := simp)]
lemma pOpcycles_opcyclesIsoSc'_hom :
    K.pOpcycles j ≫ (K.opcyclesIsoSc' i j k hi hk).hom = (K.sc' i j k).pOpcycles 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} {c : ComplexShape ι} (K : HomologicalComplex C c) (i j k : ι) (hi : c.prev j = i) (hk : c.next j = k) [K.HasHomology j] [(K.sc' i j k).HasHomology]
Goal: K.pOpcycles j ≫ (K.opcyclesIsoSc' i j k hi hk).hom = (K.sc' i j k).pOpcycles
-- HomologicalComplex.opcyclesIsoSc'_inv_fromOpcycles
@[reassoc (attr := simp)]
lemma opcyclesIsoSc'_inv_fromOpcycles :
    (K.opcyclesIsoSc' i j k hi hk).inv ≫ K.fromOpcycles j k =
      (K.sc' i j k).fromOpcycles 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} {c : ComplexShape ι} (K : HomologicalComplex C c) (i j k : ι) (hi : c.prev j = i) (hk : c.next j = k) [K.HasHomology j] [(K.sc' i j k).HasHomology]
Goal: (K.opcyclesIsoSc' i j k hi hk).inv ≫ K.fromOpcycles j k = (K.sc' i j k).fromOpcycles
-- HomologicalComplex.π_homologyIsoSc'_hom
@[reassoc (attr := simp)]
lemma π_homologyIsoSc'_hom :
    K.homologyπ j ≫ (K.homologyIsoSc' i j k hi hk).hom =
      (K.cyclesIsoSc' i j k hi hk).hom ≫ (K.sc' i j k).homologyπ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} {c : ComplexShape ι} (K : HomologicalComplex C c) (i j k : ι) (hi : c.prev j = i) (hk : c.next j = k) [K.HasHomology j] [(K.sc' i j k).HasHomology]
Goal: K.homologyπ j ≫ (K.homologyIsoSc' i j k hi hk).hom = (K.cyclesIsoSc' i j k hi hk).hom ≫ (K.sc' i j k).homologyπ
-- HomologicalComplex.π_homologyIsoSc'_inv
@[reassoc (attr := simp)]
lemma π_homologyIsoSc'_inv :
    (K.sc' i j k).homologyπ ≫ (K.homologyIsoSc' i j k hi hk).inv =
      (K.cyclesIsoSc' i j k hi hk).inv ≫ K.homologyπ j 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} {c : ComplexShape ι} (K : HomologicalComplex C c) (i j k : ι) (hi : c.prev j = i) (hk : c.next j = k) [K.HasHomology j] [(K.sc' i j k).HasHomology]
Goal: (K.sc' i j k).homologyπ ≫ (K.homologyIsoSc' i j k hi hk).inv = (K.cyclesIsoSc' i j k hi hk).inv ≫ K.homologyπ j
-- HomologicalComplex.homologyIsoSc'_hom_ι
@[reassoc (attr := simp)]
lemma homologyIsoSc'_hom_ι :
    (K.homologyIsoSc' i j k hi hk).hom ≫ (K.sc' i j k).homologyι =
      K.homologyι j ≫ (K.opcyclesIsoSc' i j k hi hk).hom 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} {c : ComplexShape ι} (K : HomologicalComplex C c) (i j k : ι) (hi : c.prev j = i) (hk : c.next j = k) [K.HasHomology j] [(K.sc' i j k).HasHomology]
Goal: (K.homologyIsoSc' i j k hi hk).hom ≫ (K.sc' i j k).homologyι = K.homologyι j ≫ (K.opcyclesIsoSc' i j k hi hk).hom
-- HomologicalComplex.homologyIsoSc'_inv_ι
@[reassoc (attr := simp)]
lemma homologyIsoSc'_inv_ι :
    (K.homologyIsoSc' i j k hi hk).inv ≫ K.homologyι j =
      (K.sc' i j k).homologyι ≫ (K.opcyclesIsoSc' i j k hi hk).inv 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} {c : ComplexShape ι} (K : HomologicalComplex C c) (i j k : ι) (hi : c.prev j = i) (hk : c.next j = k) [K.HasHomology j] [(K.sc' i j k).HasHomology]
Goal: (K.homologyIsoSc' i j k hi hk).inv ≫ K.homologyι j = (K.sc' i j k).homologyι ≫ (K.opcyclesIsoSc' i j k hi hk).inv
-- HomologicalComplex.restriction_d
@[simps]
def restriction : HomologicalComplex C c 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {C : Type u_3} [CategoryTheory.Category.{u_4, u_3} C] [CategoryTheory.Limits.HasZeroMorphisms C] (K : HomologicalComplex C c') (e : c.Embedding c') [e.IsRelIff]
Goal: ∀ (x x_1 : ι), (K.restriction e).d x x_1 = K.d (e.f x) (e.f x_1)
-- HomologicalComplex.restriction_X
@[simps]
def restriction : HomologicalComplex C c 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {C : Type u_3} [CategoryTheory.Category.{u_4, u_3} C] [CategoryTheory.Limits.HasZeroMorphisms C] (K : HomologicalComplex C c') (e : c.Embedding c') [e.IsRelIff] (i : ι)
Goal: (K.restriction e).X i = K.X (e.f i)
-- HomologicalComplex.restriction_d_eq
@[reassoc]
lemma restriction_d_eq {i j : ι} {i' j' : ι'} (hi : e.f i = i') (hj : e.f j = j') :
    (K.restriction e).d i j = (K.restrictionXIso e hi).hom ≫ K.d i' j' ≫
      (K.restrictionXIso e hj).inv 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {C : Type u_3} [CategoryTheory.Category.{u_4, u_3} C] [CategoryTheory.Limits.HasZeroMorphisms C] (K : HomologicalComplex C c') (e : c.Embedding c') [e.IsRelIff] {i j : ι} {i' j' : ι'} (hi : e.f i = i') (hj : e.f j = j')
Goal: (K.restriction e).d i j = (K.restrictionXIso e hi).hom ≫ K.d i' j' ≫ (K.restrictionXIso e hj).inv
-- HomologicalComplex.restrictionMap_f
@[simps]
def restrictionMap : K.restriction e ⟶ L.restriction e 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {C : Type u_3} [CategoryTheory.Category.{u_4, u_3} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K L : HomologicalComplex C c'} (φ : K ⟶ L) (e : c.Embedding c') [e.IsRelIff] (i : ι)
Goal: (HomologicalComplex.restrictionMap φ e).f i = φ.f (e.f i)
-- HomologicalComplex.restrictionMap_f'
@[reassoc]
lemma restrictionMap_f' {i : ι} {i' : ι'} (hi : e.f i = i') :
    (restrictionMap φ e).f i = (K.restrictionXIso e hi).hom ≫
      φ.f i' ≫ (L.restrictionXIso e hi).inv 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {C : Type u_3} [CategoryTheory.Category.{u_4, u_3} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K L : HomologicalComplex C c'} (φ : K ⟶ L) (e : c.Embedding c') [e.IsRelIff] {i : ι} {i' : ι'} (hi : e.f i = i')
Goal: (HomologicalComplex.restrictionMap φ e).f i = (K.restrictionXIso e hi).hom ≫ φ.f i' ≫ (L.restrictionXIso e hi).inv
-- HomologicalComplex.restrictionMap_id
@[simp]
lemma restrictionMap_id : restrictionMap (𝟙 K) e = 𝟙 _ 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {C : Type u_3} [CategoryTheory.Category.{u_4, u_3} C] [CategoryTheory.Limits.HasZeroMorphisms C] (K : HomologicalComplex C c') (e : c.Embedding c') [e.IsRelIff]
Goal: HomologicalComplex.restrictionMap (𝟙 K) e = 𝟙 (K.restriction e)
-- HomologicalComplex.restrictionMap_comp
@[simp, reassoc]
lemma restrictionMap_comp :
    restrictionMap (φ ≫ φ') e = restrictionMap φ e ≫ restrictionMap φ' e 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {C : Type u_3} [CategoryTheory.Category.{u_4, u_3} C] [CategoryTheory.Limits.HasZeroMorphisms C] (K : HomologicalComplex C c') {L : HomologicalComplex C c'} (M : HomologicalComplex C c') (φ : K ⟶ L) (φ' : L ⟶ M) (e : c.Embedding c') [e.IsRelIff]
Goal: HomologicalComplex.restrictionMap (φ ≫ φ') e =
  HomologicalComplex.restrictionMap φ e ≫ HomologicalComplex.restrictionMap φ' e
-- ComplexShape.Embedding.restrictionFunctor_obj
@[simps]
noncomputable def restrictionFunctor [HasZeroMorphisms C] :
    HomologicalComplex C c' ⥤ HomologicalComplex C c 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} (e : c.Embedding c') (C : Type u_3) [CategoryTheory.Category.{u_4, u_3} C] [e.IsRelIff] [CategoryTheory.Limits.HasZeroMorphisms C] (K : HomologicalComplex C c')
Goal: (e.restrictionFunctor C).obj K = K.restriction e
-- ComplexShape.Embedding.restrictionFunctor_map
@[simps]
noncomputable def restrictionFunctor [HasZeroMorphisms C] :
    HomologicalComplex C c' ⥤ HomologicalComplex C c 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} (e : c.Embedding c') (C : Type u_3) [CategoryTheory.Category.{u_4, u_3} C] [e.IsRelIff] [CategoryTheory.Limits.HasZeroMorphisms C]
Goal: ∀ {X Y : HomologicalComplex C c'} (φ : X ⟶ Y), (e.restrictionFunctor C).map φ = HomologicalComplex.restrictionMap φ e
-- CategoryTheory.ShortComplex.smul_τ₁
@[simp] lemma smul_τ₁ (a : R) (φ : S₁ ⟶ S₂) : (a • φ).τ₁ = a • φ.τ₁ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} (a : R) (φ : S₁ ⟶ S₂)
Goal: (a • φ).τ₁ = a • φ.τ₁
-- CategoryTheory.ShortComplex.smul_τ₂
@[simp] lemma smul_τ₂ (a : R) (φ : S₁ ⟶ S₂) : (a • φ).τ₂ = a • φ.τ₂ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} (a : R) (φ : S₁ ⟶ S₂)
Goal: (a • φ).τ₂ = a • φ.τ₂
-- CategoryTheory.ShortComplex.smul_τ₃
@[simp] lemma smul_τ₃ (a : R) (φ : S₁ ⟶ S₂) : (a • φ).τ₃ = a • φ.τ₃ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} (a : R) (φ : S₁ ⟶ S₂)
Goal: (a • φ).τ₃ = a • φ.τ₃
-- CategoryTheory.ShortComplex.LeftHomologyMapData.smul_φH
@[simps]
def smul (a : R) : LeftHomologyMapData (a • φ) h₁ h₂ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData} {h₂ : S₂.LeftHomologyData} (γ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂) (a : R)
Goal: (γ.smul a).φH = a • γ.φH
-- CategoryTheory.ShortComplex.LeftHomologyMapData.smul_φK
@[simps]
def smul (a : R) : LeftHomologyMapData (a • φ) h₁ h₂ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData} {h₂ : S₂.LeftHomologyData} (γ : CategoryTheory.ShortComplex.LeftHomologyMapData φ h₁ h₂) (a : R)
Goal: (γ.smul a).φK = a • γ.φK
-- CategoryTheory.ShortComplex.leftHomologyMap'_smul
@[simp]
lemma leftHomologyMap'_smul :
    leftHomologyMap' (a • φ) h₁ h₂ = a • leftHomologyMap' φ h₁ h₂ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData) (a : R)
Goal: CategoryTheory.ShortComplex.leftHomologyMap' (a • φ) h₁ h₂ = a • CategoryTheory.ShortComplex.leftHomologyMap' φ h₁ h₂
-- CategoryTheory.ShortComplex.cyclesMap'_smul
@[simp]
lemma cyclesMap'_smul :
    cyclesMap' (a • φ) h₁ h₂ = a • cyclesMap' φ h₁ h₂ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData) (a : R)
Goal: CategoryTheory.ShortComplex.cyclesMap' (a • φ) h₁ h₂ = a • CategoryTheory.ShortComplex.cyclesMap' φ h₁ h₂
-- CategoryTheory.ShortComplex.leftHomologyMap_smul
@[simp]
lemma leftHomologyMap_smul : leftHomologyMap (a • φ) = a • leftHomologyMap φ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (a : R) [S₁.HasLeftHomology] [S₂.HasLeftHomology]
Goal: CategoryTheory.ShortComplex.leftHomologyMap (a • φ) = a • CategoryTheory.ShortComplex.leftHomologyMap φ
-- CategoryTheory.ShortComplex.cyclesMap_smul
@[simp]
lemma cyclesMap_smul : cyclesMap (a • φ) = a • cyclesMap φ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (a : R) [S₁.HasLeftHomology] [S₂.HasLeftHomology]
Goal: CategoryTheory.ShortComplex.cyclesMap (a • φ) = a • CategoryTheory.ShortComplex.cyclesMap φ
-- CategoryTheory.ShortComplex.RightHomologyMapData.smul_φH
@[simps]
def smul (a : R) : RightHomologyMapData (a • φ) h₁ h₂ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} (γ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂) (a : R)
Goal: (γ.smul a).φH = a • γ.φH
-- CategoryTheory.ShortComplex.RightHomologyMapData.smul_φQ
@[simps]
def smul (a : R) : RightHomologyMapData (a • φ) h₁ h₂ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData} {h₂ : S₂.RightHomologyData} (γ : CategoryTheory.ShortComplex.RightHomologyMapData φ h₁ h₂) (a : R)
Goal: (γ.smul a).φQ = a • γ.φQ
-- CategoryTheory.ShortComplex.rightHomologyMap'_smul
@[simp]
lemma rightHomologyMap'_smul :
    rightHomologyMap' (a • φ) h₁ h₂ = a • rightHomologyMap' φ h₁ h₂ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData) (a : R)
Goal: CategoryTheory.ShortComplex.rightHomologyMap' (a • φ) h₁ h₂ = a • CategoryTheory.ShortComplex.rightHomologyMap' φ h₁ h₂
-- CategoryTheory.ShortComplex.opcyclesMap'_smul
@[simp]
lemma opcyclesMap'_smul :
    opcyclesMap' (a • φ) h₁ h₂ = a • opcyclesMap' φ h₁ h₂ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData) (a : R)
Goal: CategoryTheory.ShortComplex.opcyclesMap' (a • φ) h₁ h₂ = a • CategoryTheory.ShortComplex.opcyclesMap' φ h₁ h₂
-- CategoryTheory.ShortComplex.rightHomologyMap_smul
@[simp]
lemma rightHomologyMap_smul : rightHomologyMap (a • φ) = a • rightHomologyMap φ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (a : R) [S₁.HasRightHomology] [S₂.HasRightHomology]
Goal: CategoryTheory.ShortComplex.rightHomologyMap (a • φ) = a • CategoryTheory.ShortComplex.rightHomologyMap φ
-- CategoryTheory.ShortComplex.opcyclesMap_smul
@[simp]
lemma opcyclesMap_smul : opcyclesMap (a • φ) = a • opcyclesMap φ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (a : R) [S₁.HasRightHomology] [S₂.HasRightHomology]
Goal: CategoryTheory.ShortComplex.opcyclesMap (a • φ) = a • CategoryTheory.ShortComplex.opcyclesMap φ
-- CategoryTheory.ShortComplex.HomologyMapData.smul_left
@[simps]
def smul (a : R) : HomologyMapData (a • φ) h₁ h₂ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData} (γ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂) (a : R)
Goal: (γ.smul a).left = γ.left.smul a
-- CategoryTheory.ShortComplex.HomologyMapData.smul_right
@[simps]
def smul (a : R) : HomologyMapData (a • φ) h₁ h₂ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} {h₁ : S₁.HomologyData} {h₂ : S₂.HomologyData} (γ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂) (a : R)
Goal: (γ.smul a).right = γ.right.smul a
-- CategoryTheory.ShortComplex.homologyMap'_smul
@[simp]
lemma homologyMap'_smul :
    homologyMap' (a • φ) h₁ h₂ = a • homologyMap' φ h₁ h₂ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ : S₁ ⟶ S₂} (h₁ : S₁.HomologyData) (h₂ : S₂.HomologyData) (a : R)
Goal: CategoryTheory.ShortComplex.homologyMap' (a • φ) h₁ h₂ = a • CategoryTheory.ShortComplex.homologyMap' φ h₁ h₂
-- CategoryTheory.ShortComplex.homologyMap_smul
@[simp]
lemma homologyMap_smul [S₁.HasHomology] [S₂.HasHomology] :
    homologyMap (a • φ) = a • homologyMap φ 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) (a : R) [S₁.HasHomology] [S₂.HasHomology]
Goal: CategoryTheory.ShortComplex.homologyMap (a • φ) = a • CategoryTheory.ShortComplex.homologyMap φ
-- CategoryTheory.ShortComplex.Homotopy.smul_h₃
@[simps]
def Homotopy.smul {φ₁ φ₂ : S₁ ⟶ S₂} (h : Homotopy φ₁ φ₂) (a : R) :
    Homotopy (a • φ₁) (a • φ₂) 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (a : R)
Goal: (h.smul a).h₃ = a • h.h₃
-- CategoryTheory.ShortComplex.Homotopy.smul_h₀
@[simps]
def Homotopy.smul {φ₁ φ₂ : S₁ ⟶ S₂} (h : Homotopy φ₁ φ₂) (a : R) :
    Homotopy (a • φ₁) (a • φ₂) 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (a : R)
Goal: (h.smul a).h₀ = a • h.h₀
-- CategoryTheory.ShortComplex.Homotopy.smul_h₁
@[simps]
def Homotopy.smul {φ₁ φ₂ : S₁ ⟶ S₂} (h : Homotopy φ₁ φ₂) (a : R) :
    Homotopy (a • φ₁) (a • φ₂) 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (a : R)
Goal: (h.smul a).h₁ = a • h.h₁
-- CategoryTheory.ShortComplex.Homotopy.smul_h₂
@[simps]
def Homotopy.smul {φ₁ φ₂ : S₁ ⟶ S₂} (h : Homotopy φ₁ φ₂) (a : R) :
    Homotopy (a • φ₁) (a • φ₂) 
Arguments: {R : Type u_1} {C : Type u_2} [Semiring R] [CategoryTheory.Category.{u_3, u_2} C] [CategoryTheory.Preadditive C] [CategoryTheory.Linear R C] {S₁ S₂ : CategoryTheory.ShortComplex C} {φ₁ φ₂ : S₁ ⟶ S₂} (h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂) (a : R)
Goal: (h.smul a).h₂ = a • h.h₂
-- HomologicalComplex.truncGE'.d_comp_d
@[reassoc (attr := simp)]
lemma d_comp_d (i j k : ι) : d K e i j ≫ d K e j k = 0 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {C : Type u_3} [CategoryTheory.Category.{u_4, u_3} C] [CategoryTheory.Limits.HasZeroMorphisms C] (K : HomologicalComplex C c') (e : c.Embedding c') [e.IsTruncGE] [∀ (i' : ι'), K.HasHomology i'] (i j k : ι)
Goal: HomologicalComplex.truncGE'.d K e i j ≫ HomologicalComplex.truncGE'.d K e j k = 0
-- HomologicalComplex.truncGE'_d_eq
lemma truncGE'_d_eq {i j : ι} (hij : c.Rel i j)  {i' j' : ι'}
    (hi' : e.f i = i') (hj' : e.f j = j')  (hi : ¬ e.BoundaryGE i) :
    (K.truncGE' e).d i j = (K.truncGE'XIso e hi' hi).hom ≫ K.d i' j' ≫
      (K.truncGE'XIso e hj' (e.not_boundaryGE_next hij)).inv 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {C : Type u_3} [CategoryTheory.Category.{u_4, u_3} C] [CategoryTheory.Limits.HasZeroMorphisms C] (K : HomologicalComplex C c') (e : c.Embedding c') [e.IsTruncGE] [∀ (i' : ι'), K.HasHomology i'] {i j : ι} (hij : c.Rel i j) {i' j' : ι'} (hi' : e.f i = i') (hj' : e.f j = j') (hi : ¬e.BoundaryGE i)
Goal: (K.truncGE' e).d i j = (K.truncGE'XIso e hi' hi).hom ≫ K.d i' j' ≫ (K.truncGE'XIso e hj' ⋯).inv
-- HomologicalComplex.truncGE'_d_eq_fromOpcycles
lemma truncGE'_d_eq_fromOpcycles {i j : ι} (hij : c.Rel i j) {i' j' : ι'}
    (hi' : e.f i = i') (hj' : e.f j = j') (hi : e.BoundaryGE i) :
    (K.truncGE' e).d i j = (K.truncGE'XIsoOpcycles e hi' hi).hom ≫ K.fromOpcycles i' j' ≫
      (K.truncGE'XIso e hj' (e.not_boundaryGE_next hij)).inv 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {C : Type u_3} [CategoryTheory.Category.{u_4, u_3} C] [CategoryTheory.Limits.HasZeroMorphisms C] (K : HomologicalComplex C c') (e : c.Embedding c') [e.IsTruncGE] [∀ (i' : ι'), K.HasHomology i'] {i j : ι} (hij : c.Rel i j) {i' j' : ι'} (hi' : e.f i = i') (hj' : e.f j = j') (hi : e.BoundaryGE i)
Goal: (K.truncGE' e).d i j = (K.truncGE'XIsoOpcycles e hi' hi).hom ≫ K.fromOpcycles i' j' ≫ (K.truncGE'XIso e hj' ⋯).inv
-- ComplexShape.Embedding.boundaryGE
lemma boundaryGE {i' : ι'} {j : ι} (hj : c'.Rel i' (e.f j)) (hi' : ∀ i, e.f i ≠ i') :
    e.BoundaryGE j 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} (e : c.Embedding c') {i' : ι'} {j : ι} (hj : c'.Rel i' (e.f j)) (hi' : ∀ (i : ι), e.f i ≠ i')
Goal: e.BoundaryGE j
-- ComplexShape.Embedding.not_boundaryGE_next
lemma not_boundaryGE_next [e.IsRelIff] {j k : ι} (hk : c.Rel j k) :
    ¬ e.BoundaryGE k 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} (e : c.Embedding c') [e.IsRelIff] {j k : ι} (hk : c.Rel j k)
Goal: ¬e.BoundaryGE k
-- ComplexShape.Embedding.not_boundaryGE_next'
lemma not_boundaryGE_next' [e.IsRelIff] {j k : ι} (hj : ¬ e.BoundaryGE j) (hk : c.next j = k) :
    ¬ e.BoundaryGE k 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} (e : c.Embedding c') [e.IsRelIff] {j k : ι} (hj : ¬e.BoundaryGE j) (hk : c.next j = k)
Goal: ¬e.BoundaryGE k
-- ComplexShape.Embedding.BoundaryGE.not_mem
lemma BoundaryGE.not_mem {j : ι} (hj : e.BoundaryGE j) {i' : ι'} (hi' : c'.Rel i' (e.f j))
    (a : ι) : e.f a ≠ i' 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {e : c.Embedding c'} {j : ι} (hj : e.BoundaryGE j) {i' : ι'} (hi' : c'.Rel i' (e.f j)) (a : ι)
Goal: e.f a ≠ i'
-- ComplexShape.Embedding.prev_f_of_not_boundaryGE
lemma prev_f_of_not_boundaryGE [e.IsRelIff] {i j : ι} (hij : c.prev j = i)
    (hj : ¬ e.BoundaryGE j) :
    c'.prev (e.f j) = e.f i 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} (e : c.Embedding c') [e.IsRelIff] {i j : ι} (hij : c.prev j = i) (hj : ¬e.BoundaryGE j)
Goal: c'.prev (e.f j) = e.f i
-- ComplexShape.Embedding.BoundaryGE.false_of_isTruncLE
lemma BoundaryGE.false_of_isTruncLE {j : ι} (hj : e.BoundaryGE j) [e.IsTruncLE] : False 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} {e : c.Embedding c'} {j : ι} (hj : e.BoundaryGE j) [e.IsTruncLE]
Goal: False
-- ComplexShape.Embedding.boundaryLE
lemma boundaryLE {k' : ι'} {j : ι} (hj : c'.Rel (e.f j) k') (hk' : ∀ i, e.f i ≠ k') :
    e.BoundaryLE j 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} (e : c.Embedding c') {k' : ι'} {j : ι} (hj : c'.Rel (e.f j) k') (hk' : ∀ (i : ι), e.f i ≠ k')
Goal: e.BoundaryLE j
-- ComplexShape.Embedding.not_boundaryLE_prev
lemma not_boundaryLE_prev [e.IsRelIff] {i j : ι} (hi : c.Rel i j) :
    ¬ e.BoundaryLE i 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} (e : c.Embedding c') [e.IsRelIff] {i j : ι} (hi : c.Rel i j)
Goal: ¬e.BoundaryLE i
-- ComplexShape.Embedding.not_boundaryLE_prev'
lemma not_boundaryLE_prev' [e.IsRelIff] {i j : ι} (hj : ¬ e.BoundaryLE j) (hk : c.prev j = i) :
    ¬ e.BoundaryLE i 
Arguments: {ι : Type u_1} {ι' : Type u_2} {c : ComplexShape ι} {c' : ComplexShape ι'} (e : c.Embedding c') [e.IsRelIff] {i j : ι} (hj : ¬e.BoundaryLE j) (hk : c.prev j = i)
Goal: ¬e.BoundaryLE i