ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HomologicalComplex.opcyclesMap_zero
@[simp]
lemma opcyclesMap_zero : opcyclesMap (0 : K ⟶ L) i = 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} {c : ComplexShape ι} (K L : HomologicalComplex C c) (i : ι) [K.HasHomology i] [L.HasHomology i]
Goal: HomologicalComplex.opcyclesMap 0 i = 0
-- HomologicalComplex.homologyπ_naturality
@[reassoc (attr := simp)]
lemma homologyπ_naturality :
    K.homologyπ i ≫ homologyMap φ i = cyclesMap φ i ≫ L.homologyπ i 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} {c : ComplexShape ι} {K L : HomologicalComplex C c} (φ : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i]
Goal: K.homologyπ i ≫ HomologicalComplex.homologyMap φ i = HomologicalComplex.cyclesMap φ i ≫ L.homologyπ i
-- HomologicalComplex.homologyι_naturality
@[reassoc (attr := simp)]
lemma homologyι_naturality :
    homologyMap φ i ≫ L.homologyι i = K.homologyι i ≫ opcyclesMap φ i 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} {c : ComplexShape ι} {K L : HomologicalComplex C c} (φ : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i]
Goal: HomologicalComplex.homologyMap φ i ≫ L.homologyι i = K.homologyι i ≫ HomologicalComplex.opcyclesMap φ i
-- HomologicalComplex.homology_π_ι
@[reassoc (attr := simp)]
lemma homology_π_ι :
    K.homologyπ i ≫ K.homologyι i = K.iCycles i ≫ K.pOpcycles i 
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 : ι) [K.HasHomology i]
Goal: K.homologyπ i ≫ K.homologyι i = K.iCycles i ≫ K.pOpcycles i
-- HomologicalComplex.opcyclesMap_comp_descOpcycles
@[reassoc (attr := simp)]
lemma opcyclesMap_comp_descOpcycles {A : C} (k : L.X i ⟶ A) (j : ι) (hj : c.prev i = j)
    (hk : L.d j i ≫ k = 0) (φ : K ⟶ L) :
    opcyclesMap φ i ≫ L.descOpcycles k j hj hk = K.descOpcycles (φ.f i ≫ k) j hj
      (by rw [← φ.comm_assoc, hk, comp_zero]) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} {c : ComplexShape ι} {K L : HomologicalComplex C c} {i : ι} [K.HasHomology i] [L.HasHomology i] {A : C} (k : L.X i ⟶ A) (j : ι) (hj : c.prev i = j) (hk : L.d j i ≫ k = 0) (φ : K ⟶ L)
Goal: HomologicalComplex.opcyclesMap φ i ≫ L.descOpcycles k j hj hk = K.descOpcycles (φ.f i ≫ k) j hj ⋯
-- HomologicalComplex.liftCycles_comp_cyclesMap
@[reassoc (attr := simp)]
lemma liftCycles_comp_cyclesMap {A : C} (k : A ⟶ K.X i) (j : ι) (hj : c.next i = j)
    (hk : k ≫ K.d i j = 0) (φ : K ⟶ L) :
    K.liftCycles k j hj hk ≫ cyclesMap φ i = L.liftCycles (k ≫ φ.f i) j hj
      (by rw [assoc, φ.comm, reassoc_of% hk, zero_comp]) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} {c : ComplexShape ι} {K L : HomologicalComplex C c} {i : ι} [K.HasHomology i] [L.HasHomology i] {A : C} (k : A ⟶ K.X i) (j : ι) (hj : c.next i = j) (hk : k ≫ K.d i j = 0) (φ : K ⟶ L)
Goal: K.liftCycles k j hj hk ≫ HomologicalComplex.cyclesMap φ i = L.liftCycles (k ≫ φ.f i) j hj ⋯
-- HomologicalComplex.homologyFunctor_map
@[simps]
noncomputable def homologyFunctor [CategoryWithHomology C] : HomologicalComplex C c ⥤ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} (c : ComplexShape ι) (i : ι) [CategoryTheory.CategoryWithHomology C]
Goal: ∀ {X Y : HomologicalComplex C c} (f : X ⟶ Y),
  (HomologicalComplex.homologyFunctor C c i).map f = HomologicalComplex.homologyMap f i
-- HomologicalComplex.homologyFunctor_obj
@[simps]
noncomputable def homologyFunctor [CategoryWithHomology C] : HomologicalComplex C c ⥤ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} (c : ComplexShape ι) (i : ι) [CategoryTheory.CategoryWithHomology C] (K : HomologicalComplex C c)
Goal: (HomologicalComplex.homologyFunctor C c i).obj K = K.homology i
-- HomologicalComplex.gradedHomologyFunctor_obj
@[simps]
noncomputable def gradedHomologyFunctor [CategoryWithHomology C] :
    HomologicalComplex C c ⥤ GradedObject ι C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} (c : ComplexShape ι) [CategoryTheory.CategoryWithHomology C] (K : HomologicalComplex C c) (i : ι)
Goal: (HomologicalComplex.gradedHomologyFunctor C c).obj K i = K.homology i
-- HomologicalComplex.gradedHomologyFunctor_map
@[simps]
noncomputable def gradedHomologyFunctor [CategoryWithHomology C] :
    HomologicalComplex C c ⥤ GradedObject ι C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} (c : ComplexShape ι) [CategoryTheory.CategoryWithHomology C]
Goal: ∀ {X Y : HomologicalComplex C c} (f : X ⟶ Y) (i : ι),
  (HomologicalComplex.gradedHomologyFunctor C c).map f i = HomologicalComplex.homologyMap f i
-- HomologicalComplex.cyclesFunctor_obj
@[simps]
noncomputable def cyclesFunctor [CategoryWithHomology C] : HomologicalComplex C c ⥤ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} (c : ComplexShape ι) (i : ι) [CategoryTheory.CategoryWithHomology C] (K : HomologicalComplex C c)
Goal: (HomologicalComplex.cyclesFunctor C c i).obj K = K.cycles i
-- HomologicalComplex.cyclesFunctor_map
@[simps]
noncomputable def cyclesFunctor [CategoryWithHomology C] : HomologicalComplex C c ⥤ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} (c : ComplexShape ι) (i : ι) [CategoryTheory.CategoryWithHomology C]
Goal: ∀ {X Y : HomologicalComplex C c} (f : X ⟶ Y),
  (HomologicalComplex.cyclesFunctor C c i).map f = HomologicalComplex.cyclesMap f i
-- HomologicalComplex.opcyclesFunctor_map
@[simps]
noncomputable def opcyclesFunctor [CategoryWithHomology C] : HomologicalComplex C c ⥤ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} (c : ComplexShape ι) (i : ι) [CategoryTheory.CategoryWithHomology C]
Goal: ∀ {X Y : HomologicalComplex C c} (f : X ⟶ Y),
  (HomologicalComplex.opcyclesFunctor C c i).map f = HomologicalComplex.opcyclesMap f i
-- HomologicalComplex.opcyclesFunctor_obj
@[simps]
noncomputable def opcyclesFunctor [CategoryWithHomology C] : HomologicalComplex C c ⥤ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} (c : ComplexShape ι) (i : ι) [CategoryTheory.CategoryWithHomology C] (K : HomologicalComplex C c)
Goal: (HomologicalComplex.opcyclesFunctor C c i).obj K = K.opcycles i
-- HomologicalComplex.natTransHomologyπ_app
@[simps]
noncomputable def natTransHomologyπ [CategoryWithHomology C] :
    cyclesFunctor C c i ⟶ homologyFunctor C c i 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} (c : ComplexShape ι) (i : ι) [CategoryTheory.CategoryWithHomology C] (K : HomologicalComplex C c)
Goal: (HomologicalComplex.natTransHomologyπ C c i).app K = K.homologyπ i
-- HomologicalComplex.natTransHomologyι_app
@[simps]
noncomputable def natTransHomologyι [CategoryWithHomology C] :
    homologyFunctor C c i ⟶ opcyclesFunctor C c i 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} (c : ComplexShape ι) (i : ι) [CategoryTheory.CategoryWithHomology C] (K : HomologicalComplex C c)
Goal: (HomologicalComplex.natTransHomologyι C c i).app K = K.homologyι i
-- HomologicalComplex.homologyFunctorIso_inv_app
@[simps!]
noncomputable def homologyFunctorIso [CategoryWithHomology C] :
    homologyFunctor C c i ≅
      shortComplexFunctor C c i ⋙ ShortComplex.homologyFunctor C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} (c : ComplexShape ι) (i : ι) [CategoryTheory.CategoryWithHomology C] (X : HomologicalComplex C c)
Goal: (HomologicalComplex.homologyFunctorIso C c i).inv.app X = 𝟙 (X.homology i)
-- HomologicalComplex.homologyFunctorIso_hom_app
@[simps!]
noncomputable def homologyFunctorIso [CategoryWithHomology C] :
    homologyFunctor C c i ≅
      shortComplexFunctor C c i ⋙ ShortComplex.homologyFunctor C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} (c : ComplexShape ι) (i : ι) [CategoryTheory.CategoryWithHomology C] (X : HomologicalComplex C c)
Goal: (HomologicalComplex.homologyFunctorIso C c i).hom.app X = 𝟙 (X.homology i)
-- HomologicalComplex.isIso_iCycles
lemma isIso_iCycles : IsIso (K.iCycles i) 
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 : ι) (hj : c.next i = j) (h : K.d i j = 0) [K.HasHomology i]
Goal: CategoryTheory.IsIso (K.iCycles i)
-- HomologicalComplex.iCyclesIso_hom
@[simps! hom]
noncomputable def iCyclesIso : K.cycles i ≅ K.X i 
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 : ι) (hj : c.next i = j) (h : K.d i j = 0) [K.HasHomology i]
Goal: (K.iCyclesIso i j hj h).hom = K.iCycles i
-- HomologicalComplex.iCyclesIso_hom_inv_id
@[reassoc (attr := simp)]
lemma iCyclesIso_hom_inv_id :
    K.iCycles i ≫ (K.iCyclesIso i j hj h).inv = 𝟙 _ 
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 : ι) (hj : c.next i = j) (h : K.d i j = 0) [K.HasHomology i]
Goal: K.iCycles i ≫ (K.iCyclesIso i j hj h).inv = 𝟙 (K.cycles i)
-- HomologicalComplex.iCyclesIso_inv_hom_id
@[reassoc (attr := simp)]
lemma iCyclesIso_inv_hom_id :
    (K.iCyclesIso i j hj h).inv ≫ K.iCycles i = 𝟙 _ 
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 : ι) (hj : c.next i = j) (h : K.d i j = 0) [K.HasHomology i]
Goal: (K.iCyclesIso i j hj h).inv ≫ K.iCycles i = 𝟙 (K.X i)
-- HomologicalComplex.isIso_homologyι
lemma isIso_homologyι : IsIso (K.homologyι i) 
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 : ι) (hj : c.next i = j) (h : K.d i j = 0) [K.HasHomology i]
Goal: CategoryTheory.IsIso (K.homologyι i)
-- HomologicalComplex.isoHomologyι_hom
@[simps! hom]
noncomputable def isoHomologyι : K.homology i ≅ K.opcycles i 
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 : ι) (hj : c.next i = j) (h : K.d i j = 0) [K.HasHomology i]
Goal: (K.isoHomologyι i j hj h).hom = K.homologyι i
-- HomologicalComplex.isoHomologyι_hom_inv_id
@[reassoc (attr := simp)]
lemma isoHomologyι_hom_inv_id :
    K.homologyι i ≫ (K.isoHomologyι i j hj h).inv = 𝟙 _ 
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 : ι) (hj : c.next i = j) (h : K.d i j = 0) [K.HasHomology i]
Goal: K.homologyι i ≫ (K.isoHomologyι i j hj h).inv = 𝟙 (K.homology i)
-- HomologicalComplex.isoHomologyι_inv_hom_id
@[reassoc (attr := simp)]
lemma isoHomologyι_inv_hom_id :
    (K.isoHomologyι i j hj h).inv ≫ K.homologyι i = 𝟙 _ 
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 : ι) (hj : c.next i = j) (h : K.d i j = 0) [K.HasHomology i]
Goal: (K.isoHomologyι i j hj h).inv ≫ K.homologyι i = 𝟙 (K.opcycles i)
-- HomologicalComplex.isIso_pOpcycles
lemma isIso_pOpcycles : IsIso (K.pOpcycles 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 : ι) (hi : c.prev j = i) (h : K.d i j = 0) [K.HasHomology j]
Goal: CategoryTheory.IsIso (K.pOpcycles j)
-- HomologicalComplex.pOpcyclesIso_hom
@[simps! hom]
noncomputable def pOpcyclesIso : K.X j ≅ K.opcycles 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 : ι) (hi : c.prev j = i) (h : K.d i j = 0) [K.HasHomology j]
Goal: (K.pOpcyclesIso i j hi h).hom = K.pOpcycles j
-- HomologicalComplex.pOpcyclesIso_hom_inv_id
@[reassoc (attr := simp)]
lemma pOpcyclesIso_hom_inv_id :
    K.pOpcycles j ≫ (K.pOpcyclesIso i j hi h).inv = 𝟙 _ 
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 : ι) (hi : c.prev j = i) (h : K.d i j = 0) [K.HasHomology j]
Goal: K.pOpcycles j ≫ (K.pOpcyclesIso i j hi h).inv = 𝟙 (K.X j)
-- HomologicalComplex.pOpcyclesIso_inv_hom_id
@[reassoc (attr := simp)]
lemma pOpcyclesIso_inv_hom_id :
    (K.pOpcyclesIso i j hi h).inv ≫ K.pOpcycles j = 𝟙 _ 
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 : ι) (hi : c.prev j = i) (h : K.d i j = 0) [K.HasHomology j]
Goal: (K.pOpcyclesIso i j hi h).inv ≫ K.pOpcycles j = 𝟙 (K.opcycles j)
-- HomologicalComplex.isIso_homologyπ
lemma isIso_homologyπ : IsIso (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 : ι) (hi : c.prev j = i) (h : K.d i j = 0) [K.HasHomology j]
Goal: CategoryTheory.IsIso (K.homologyπ j)
-- HomologicalComplex.isoHomologyπ_hom
@[simps! hom]
noncomputable def isoHomologyπ : K.cycles j ≅ K.homology 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 : ι) (hi : c.prev j = i) (h : K.d i j = 0) [K.HasHomology j]
Goal: (K.isoHomologyπ i j hi h).hom = K.homologyπ j
-- HomologicalComplex.isoHomologyπ_hom_inv_id
@[reassoc (attr := simp)]
lemma isoHomologyπ_hom_inv_id :
    K.homologyπ j ≫ (K.isoHomologyπ i j hi h).inv = 𝟙 _ 
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 : ι) (hi : c.prev j = i) (h : K.d i j = 0) [K.HasHomology j]
Goal: K.homologyπ j ≫ (K.isoHomologyπ i j hi h).inv = 𝟙 (K.cycles j)
-- HomologicalComplex.isoHomologyπ_inv_hom_id
@[reassoc (attr := simp)]
lemma isoHomologyπ_inv_hom_id :
    (K.isoHomologyπ i j hi h).inv ≫ K.homologyπ j = 𝟙 _ 
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 : ι) (hi : c.prev j = i) (h : K.d i j = 0) [K.HasHomology j]
Goal: (K.isoHomologyπ i j hi h).inv ≫ K.homologyπ j = 𝟙 (K.homology j)
-- HomologicalComplex.epi_homologyMap_of_epi_of_not_rel
lemma epi_homologyMap_of_epi_of_not_rel (φ : K ⟶ L) (i : ι)
    [K.HasHomology i] [L.HasHomology i] [Epi (φ.f i)] (hi : ∀ j, ¬ c.Rel i j) :
    Epi (homologyMap φ i) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {ι : Type u_2} {c : ComplexShape ι} {K L : HomologicalComplex C c} (φ : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i] [CategoryTheory.Epi (φ.f i)] (hi : ∀ (j : ι), ¬c.Rel i j)
Goal: CategoryTheory.Epi (HomologicalComplex.homologyMap φ i)
-- HomologicalComplex.mono_homologyMap_of_mono_of_not_rel
lemma mono_homologyMap_of_mono_of_not_rel (φ : K ⟶ L) (j : ι)
    [K.HasHomology j] [L.HasHomology j] [Mono (φ.f j)] (hj : ∀ i, ¬ c.Rel i j) :
    Mono (homologyMap φ 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 L : HomologicalComplex C c} (φ : K ⟶ L) (j : ι) [K.HasHomology j] [L.HasHomology j] [CategoryTheory.Mono (φ.f j)] (hj : ∀ (i : ι), ¬c.Rel i j)
Goal: CategoryTheory.Mono (HomologicalComplex.homologyMap φ j)
-- HomologicalComplex.exactAt_iff
lemma exactAt_iff :
    K.ExactAt i ↔ (K.sc i).Exact 
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 : ι)
Goal: K.ExactAt i ↔ (K.sc i).Exact
-- HomologicalComplex.ExactAt.of_iso
lemma ExactAt.of_iso (hK : K.ExactAt i) {L : HomologicalComplex C c} (e : K ≅ L) :
    L.ExactAt i 
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 : ι} (hK : K.ExactAt i) {L : HomologicalComplex C c} (e : K ≅ L)
Goal: L.ExactAt i
-- HomologicalComplex.exactAt_iff'
lemma exactAt_iff' (hi : c.prev j = i) (hk : c.next j = k) :
    K.ExactAt j ↔ (K.sc' i j k).Exact 
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)
Goal: K.ExactAt j ↔ (K.sc' i j k).Exact
-- HomologicalComplex.exactAt_iff_isZero_homology
lemma exactAt_iff_isZero_homology [K.HasHomology i] :
    K.ExactAt i ↔ IsZero (K.homology i) 
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 : ι) [K.HasHomology i]
Goal: K.ExactAt i ↔ CategoryTheory.Limits.IsZero (K.homology i)
-- ChainComplex.isoHomologyι₀_inv_naturality
@[reassoc (attr := simp)]
lemma isoHomologyι₀_inv_naturality [L.HasHomology 0] :
    K.isoHomologyι₀.inv ≫ HomologicalComplex.homologyMap φ 0 =
      HomologicalComplex.opcyclesMap φ 0 ≫ L.isoHomologyι₀.inv 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K L : ChainComplex C ℕ} (φ : K ⟶ L) [HomologicalComplex.HasHomology K 0] [HomologicalComplex.HasHomology L 0]
Goal: K.isoHomologyι₀.inv ≫ HomologicalComplex.homologyMap φ 0 = HomologicalComplex.opcyclesMap φ 0 ≫ L.isoHomologyι₀.inv
-- CochainComplex.isoHomologyπ₀_inv_naturality
@[reassoc (attr := simp)]
lemma isoHomologyπ₀_inv_naturality [L.HasHomology 0] :
    HomologicalComplex.homologyMap φ 0 ≫ L.isoHomologyπ₀.inv =
      K.isoHomologyπ₀.inv ≫ HomologicalComplex.cyclesMap φ 0 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Limits.HasZeroMorphisms C] {K L : CochainComplex C ℕ} (φ : K ⟶ L) [HomologicalComplex.HasHomology K 0] [HomologicalComplex.HasHomology L 0]
Goal: HomologicalComplex.homologyMap φ 0 ≫ L.isoHomologyπ₀.inv = K.isoHomologyπ₀.inv ≫ HomologicalComplex.cyclesMap φ 0
-- HomologicalComplex.homologyMap_neg
@[simp]
lemma homologyMap_neg : homologyMap (-φ) i = -homologyMap φ i 
Arguments: {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {c : ComplexShape ι} {K L : HomologicalComplex C c} (φ : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i]
Goal: HomologicalComplex.homologyMap (-φ) i = -HomologicalComplex.homologyMap φ i
-- HomologicalComplex.homologyMap_add
@[simp]
lemma homologyMap_add : homologyMap (φ + ψ) i = homologyMap φ i + homologyMap ψ i 
Arguments: {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {c : ComplexShape ι} {K L : HomologicalComplex C c} (φ ψ : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i]
Goal: HomologicalComplex.homologyMap (φ + ψ) i = HomologicalComplex.homologyMap φ i + HomologicalComplex.homologyMap ψ i
-- HomologicalComplex.homologyMap_sub
@[simp]
lemma homologyMap_sub : homologyMap (φ - ψ) i = homologyMap φ i - homologyMap ψ i 
Arguments: {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {c : ComplexShape ι} {K L : HomologicalComplex C c} (φ ψ : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i]
Goal: HomologicalComplex.homologyMap (φ - ψ) i = HomologicalComplex.homologyMap φ i - HomologicalComplex.homologyMap ψ i
-- CochainComplex.isIso_liftCycles_iff
lemma isIso_liftCycles_iff (K : CochainComplex C ℕ) {X : C} (φ : X ⟶ K.X 0)
    [K.HasHomology 0] (hφ : φ ≫ K.d 0 1 = 0) :
    IsIso (K.liftCycles φ 1 (by simp) hφ) ↔
      (ShortComplex.mk _ _ hφ).Exact ∧ Mono φ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Abelian C] (K : CochainComplex C ℕ) {X : C} (φ : X ⟶ K.X 0) [HomologicalComplex.HasHomology K 0] (hφ : φ ≫ K.d 0 1 = 0)
Goal: CategoryTheory.IsIso (HomologicalComplex.liftCycles K φ 1 ⋯ hφ) ↔
  (CategoryTheory.ShortComplex.mk φ (K.d 0 1) hφ).Exact ∧ CategoryTheory.Mono φ
-- ChainComplex.isIso_descOpcycles_iff
lemma isIso_descOpcycles_iff (K : ChainComplex C ℕ) {X : C} (φ : K.X 0 ⟶ X)
    [K.HasHomology 0] (hφ : K.d 1 0 ≫ φ = 0) :
    IsIso (K.descOpcycles φ 1 (by simp) hφ) ↔
      (ShortComplex.mk _ _ hφ).Exact ∧ Epi φ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Abelian C] (K : ChainComplex C ℕ) {X : C} (φ : K.X 0 ⟶ X) [HomologicalComplex.HasHomology K 0] (hφ : K.d 1 0 ≫ φ = 0)
Goal: CategoryTheory.IsIso (HomologicalComplex.descOpcycles K φ 1 ⋯ hφ) ↔
  (CategoryTheory.ShortComplex.mk (K.d 1 0) φ hφ).Exact ∧ CategoryTheory.Epi φ
-- HomologicalComplex.cyclesIsoSc'_hom_iCycles
@[reassoc (attr := simp)]
lemma cyclesIsoSc'_hom_iCycles :
    (K.cyclesIsoSc' i j k hi hk).hom ≫ (K.sc' i j k).iCycles = K.iCycles 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.cyclesIsoSc' i j k hi hk).hom ≫ (K.sc' i j k).iCycles = K.iCycles j
-- HomologicalComplex.cyclesIsoSc'_inv_iCycles
@[reassoc (attr := simp)]
lemma cyclesIsoSc'_inv_iCycles :
    (K.cyclesIsoSc' i j k hi hk).inv ≫ K.iCycles j = (K.sc' i j k).iCycles 
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.cyclesIsoSc' i j k hi hk).inv ≫ K.iCycles j = (K.sc' i j k).iCycles
-- HomologicalComplex.toCycles_cyclesIsoSc'_hom
@[reassoc (attr := simp)]
lemma toCycles_cyclesIsoSc'_hom :
    K.toCycles i j ≫ (K.cyclesIsoSc' i j k hi hk).hom = (K.sc' i j k).toCycles 
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.toCycles i j ≫ (K.cyclesIsoSc' i j k hi hk).hom = (K.sc' i j k).toCycles
-- HomologicalComplex.pOpcycles_opcyclesIsoSc'_inv
@[reassoc (attr := simp)]
lemma pOpcycles_opcyclesIsoSc'_inv :
    (K.sc' i j k).pOpcycles ≫ (K.opcyclesIsoSc' i j k hi hk).inv = K.pOpcycles 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).pOpcycles ≫ (K.opcyclesIsoSc' i j k hi hk).inv = K.pOpcycles j