ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- DirectSum.isInternal_submodule_iff_independent_and_iSup_eq_top
theorem isInternal_submodule_iff_independent_and_iSup_eq_top (A : ι → Submodule R M) :
    IsInternal A ↔ CompleteLattice.Independent A ∧ iSup A = ⊤ 
Arguments: {R : Type u} [Ring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommGroup M] [Module R M] (A : ι → Submodule R M)
Goal: DirectSum.IsInternal A ↔ CompleteLattice.Independent A ∧ iSup A = ⊤
-- DirectSum.isInternal_submodule_iff_isCompl
theorem isInternal_submodule_iff_isCompl (A : ι → Submodule R M) {i j : ι} (hij : i ≠ j)
    (h : (Set.univ : Set ι) = {i, j}) : IsInternal A ↔ IsCompl (A i) (A j) 
Arguments: {R : Type u} [Ring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommGroup M] [Module R M] (A : ι → Submodule R M) {i j : ι} (hij : i ≠ j) (h : Set.univ = {i, j})
Goal: DirectSum.IsInternal A ↔ IsCompl (A i) (A j)
-- DirectSum.isInternal_ne_bot_iff
@[simp]
theorem isInternal_ne_bot_iff {A : ι → Submodule R M} :
    IsInternal (fun i : {i // A i ≠ ⊥} ↦ A i) ↔ IsInternal A 
Arguments: {R : Type u} [Ring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommGroup M] [Module R M] {A : ι → Submodule R M}
Goal: (DirectSum.IsInternal fun i => A ↑i) ↔ DirectSum.IsInternal A
-- DirectSum.isInternal_biSup_submodule_of_independent
lemma isInternal_biSup_submodule_of_independent {A : ι → Submodule R M} (s : Set ι)
    (h : CompleteLattice.Independent <| fun i : s ↦ A i) :
    IsInternal <| fun (i : s) ↦ (A i).comap (⨆ i ∈ s, A i).subtype 
Arguments: {R : Type u} [Ring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommGroup M] [Module R M] {A : ι → Submodule R M} (s : Set ι) (h : CompleteLattice.Independent fun i => A ↑i)
Goal: DirectSum.IsInternal fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)
-- DirectSum.IsInternal.addSubmonoid_independent
theorem IsInternal.addSubmonoid_independent {M : Type*} [AddCommMonoid M] {A : ι → AddSubmonoid M}
    (h : IsInternal A) : CompleteLattice.Independent A 
Arguments: {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_2} [AddCommMonoid M] {A : ι → AddSubmonoid M} (h : DirectSum.IsInternal A)
Goal: CompleteLattice.Independent A
-- DirectSum.IsInternal.addSubgroup_independent
theorem IsInternal.addSubgroup_independent {M : Type*} [AddCommGroup M] {A : ι → AddSubgroup M}
    (h : IsInternal A) : CompleteLattice.Independent A 
Arguments: {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_2} [AddCommGroup M] {A : ι → AddSubgroup M} (h : DirectSum.IsInternal A)
Goal: CompleteLattice.Independent A
-- HomologicalComplex₂.shiftFunctor₁XXIso_refl
@[simp]
lemma shiftFunctor₁XXIso_refl (a b x : ℤ) :
    K.shiftFunctor₁XXIso a x (a + x) rfl b = Iso.refl _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (K : HomologicalComplex₂ C (ComplexShape.up ℤ) (ComplexShape.up ℤ)) (a b x : ℤ)
Goal: K.shiftFunctor₁XXIso a x (a + x) ⋯ b =
  CategoryTheory.Iso.refl ((((HomologicalComplex₂.shiftFunctor₁ C x).obj K).X a).X b)
-- HomologicalComplex₂.shiftFunctor₂XXIso_refl
@[simp]
lemma shiftFunctor₂XXIso_refl (a b y : ℤ) :
    K.shiftFunctor₂XXIso a b y (b + y) rfl = Iso.refl _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (K : HomologicalComplex₂ C (ComplexShape.up ℤ) (ComplexShape.up ℤ)) (a b y : ℤ)
Goal: K.shiftFunctor₂XXIso a b y (b + y) ⋯ =
  CategoryTheory.Iso.refl ((((HomologicalComplex₂.shiftFunctor₂ C y).obj K).X a).X b)
-- HomologicalComplex₂.D₁_totalShift₁XIso_hom
@[reassoc]
lemma D₁_totalShift₁XIso_hom (n₀ n₁ n₀' n₁' : ℤ) (h₀ : n₀ + x = n₀') (h₁ : n₁ + x = n₁') :
    ((shiftFunctor₁ C x).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
      x.negOnePow • ((K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁') 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (K : HomologicalComplex₂ C (ComplexShape.up ℤ) (ComplexShape.up ℤ)) (x : ℤ) [K.HasTotal (ComplexShape.up ℤ)] (n₀ n₁ n₀' n₁' : ℤ) (h₀ : n₀ + x = n₀') (h₁ : n₁ + x = n₁')
Goal: ((HomologicalComplex₂.shiftFunctor₁ C x).obj K).D₁ (ComplexShape.up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
  x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₁ (ComplexShape.up ℤ) n₀' n₁'
-- HomologicalComplex₂.D₂_totalShift₁XIso_hom
@[reassoc]
lemma D₂_totalShift₁XIso_hom (n₀ n₁ n₀' n₁' : ℤ) (h₀ : n₀ + x = n₀') (h₁ : n₁ + x = n₁') :
    ((shiftFunctor₁ C x).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
      x.negOnePow • ((K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁') 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (K : HomologicalComplex₂ C (ComplexShape.up ℤ) (ComplexShape.up ℤ)) (x : ℤ) [K.HasTotal (ComplexShape.up ℤ)] (n₀ n₁ n₀' n₁' : ℤ) (h₀ : n₀ + x = n₀') (h₁ : n₁ + x = n₁')
Goal: ((HomologicalComplex₂.shiftFunctor₁ C x).obj K).D₂ (ComplexShape.up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
  x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₂ (ComplexShape.up ℤ) n₀' n₁'
-- HomologicalComplex₂.ι_totalShift₁Iso_hom_f
@[reassoc]
lemma ι_totalShift₁Iso_hom_f (a b n : ℤ) (h : a + b = n) (a' : ℤ) (ha' : a' = a + x)
    (n' : ℤ) (hn' : n' = n + x) :
    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) a b n h ≫ (K.totalShift₁Iso x).hom.f n =
      (K.shiftFunctor₁XXIso a x a' ha' b).hom ≫ K.ιTotal (up ℤ) a' b n' (by dsimp; omega) ≫
        (CochainComplex.shiftFunctorObjXIso (K.total (up ℤ)) x n n' hn').inv 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (K : HomologicalComplex₂ C (ComplexShape.up ℤ) (ComplexShape.up ℤ)) (x : ℤ) [K.HasTotal (ComplexShape.up ℤ)] (a b n : ℤ) (h : a + b = n) (a' : ℤ) (ha' : a' = a + x) (n' : ℤ) (hn' : n' = n + x)
Goal: ((HomologicalComplex₂.shiftFunctor₁ C x).obj K).ιTotal (ComplexShape.up ℤ) a b n h ≫ (K.totalShift₁Iso x).hom.f n =
  (K.shiftFunctor₁XXIso a x a' ha' b).hom ≫
    K.ιTotal (ComplexShape.up ℤ) a' b n' ⋯ ≫
      (CochainComplex.shiftFunctorObjXIso (K.total (ComplexShape.up ℤ)) x n n' hn').inv
-- HomologicalComplex₂.ι_totalShift₁Iso_inv_f
@[reassoc]
lemma ι_totalShift₁Iso_inv_f (a b n : ℤ) (h : a + b = n) (a' n' : ℤ)
    (ha' : a' + b = n') (hn' : n' = n + x) :
    K.ιTotal (up ℤ) a' b n' ha' ≫
      (CochainComplex.shiftFunctorObjXIso (K.total (up ℤ)) x n n' hn').inv ≫
        (K.totalShift₁Iso x).inv.f n =
      (K.shiftFunctor₁XXIso a x a' (by omega) b).inv ≫
        ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) a b n h 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (K : HomologicalComplex₂ C (ComplexShape.up ℤ) (ComplexShape.up ℤ)) (x : ℤ) [K.HasTotal (ComplexShape.up ℤ)] (a b n : ℤ) (h : a + b = n) (a' n' : ℤ) (ha' : a' + b = n') (hn' : n' = n + x)
Goal: K.ιTotal (ComplexShape.up ℤ) a' b n' ha' ≫
    (CochainComplex.shiftFunctorObjXIso (K.total (ComplexShape.up ℤ)) x n n' hn').inv ≫ (K.totalShift₁Iso x).inv.f n =
  (K.shiftFunctor₁XXIso a x a' ⋯ b).inv ≫
    ((HomologicalComplex₂.shiftFunctor₁ C x).obj K).ιTotal (ComplexShape.up ℤ) a b n h
-- HomologicalComplex₂.totalShift₁Iso_hom_naturality
@[reassoc]
lemma totalShift₁Iso_hom_naturality [L.HasTotal (up ℤ)] :
    total.map ((shiftFunctor₁ C x).map f) (up ℤ) ≫ (L.totalShift₁Iso x).hom =
      (K.totalShift₁Iso x).hom ≫ (total.map f (up ℤ))⟦x⟧' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {K L : HomologicalComplex₂ C (ComplexShape.up ℤ) (ComplexShape.up ℤ)} (f : K ⟶ L) (x : ℤ) [K.HasTotal (ComplexShape.up ℤ)] [L.HasTotal (ComplexShape.up ℤ)]
Goal: HomologicalComplex₂.total.map ((HomologicalComplex₂.shiftFunctor₁ C x).map f) (ComplexShape.up ℤ) ≫
    (L.totalShift₁Iso x).hom =
  (K.totalShift₁Iso x).hom ≫
    (CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) x).map
      (HomologicalComplex₂.total.map f (ComplexShape.up ℤ))
-- HomologicalComplex₂.D₁_totalShift₂XIso_hom
@[reassoc]
lemma D₁_totalShift₂XIso_hom (n₀ n₁ n₀' n₁' : ℤ) (h₀ : n₀ + y = n₀') (h₁ : n₁ + y = n₁') :
    ((shiftFunctor₂ C y).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
      y.negOnePow • ((K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁') 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (K : HomologicalComplex₂ C (ComplexShape.up ℤ) (ComplexShape.up ℤ)) (y : ℤ) [K.HasTotal (ComplexShape.up ℤ)] (n₀ n₁ n₀' n₁' : ℤ) (h₀ : n₀ + y = n₀') (h₁ : n₁ + y = n₁')
Goal: ((HomologicalComplex₂.shiftFunctor₂ C y).obj K).D₁ (ComplexShape.up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
  y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₁ (ComplexShape.up ℤ) n₀' n₁'
-- HomologicalComplex₂.D₂_totalShift₂XIso_hom
@[reassoc]
lemma D₂_totalShift₂XIso_hom (n₀ n₁ n₀' n₁' : ℤ) (h₀ : n₀ + y = n₀') (h₁ : n₁ + y = n₁') :
    ((shiftFunctor₂ C y).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
      y.negOnePow • ((K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁') 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (K : HomologicalComplex₂ C (ComplexShape.up ℤ) (ComplexShape.up ℤ)) (y : ℤ) [K.HasTotal (ComplexShape.up ℤ)] (n₀ n₁ n₀' n₁' : ℤ) (h₀ : n₀ + y = n₀') (h₁ : n₁ + y = n₁')
Goal: ((HomologicalComplex₂.shiftFunctor₂ C y).obj K).D₂ (ComplexShape.up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
  y.negOnePow • (K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₂ (ComplexShape.up ℤ) n₀' n₁'
-- HomologicalComplex₂.ι_totalShift₂Iso_hom_f
@[reassoc]
lemma ι_totalShift₂Iso_hom_f (a b n : ℤ) (h : a + b = n) (b' : ℤ) (hb' : b' = b + y)
    (n' : ℤ) (hn' : n' = n + y) :
    ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) a b n h ≫ (K.totalShift₂Iso y).hom.f n =
      (a * y).negOnePow • (K.shiftFunctor₂XXIso a b y b' hb').hom ≫
        K.ιTotal (up ℤ) a b' n' (by dsimp; omega) ≫
          (CochainComplex.shiftFunctorObjXIso (K.total (up ℤ)) y n n' hn').inv 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (K : HomologicalComplex₂ C (ComplexShape.up ℤ) (ComplexShape.up ℤ)) (y : ℤ) [K.HasTotal (ComplexShape.up ℤ)] (a b n : ℤ) (h : a + b = n) (b' : ℤ) (hb' : b' = b + y) (n' : ℤ) (hn' : n' = n + y)
Goal: ((HomologicalComplex₂.shiftFunctor₂ C y).obj K).ιTotal (ComplexShape.up ℤ) a b n h ≫ (K.totalShift₂Iso y).hom.f n =
  (a * y).negOnePow •
    (K.shiftFunctor₂XXIso a b y b' hb').hom ≫
      K.ιTotal (ComplexShape.up ℤ) a b' n' ⋯ ≫
        (CochainComplex.shiftFunctorObjXIso (K.total (ComplexShape.up ℤ)) y n n' hn').inv
-- HomologicalComplex₂.ι_totalShift₂Iso_inv_f
@[reassoc]
lemma ι_totalShift₂Iso_inv_f (a b n : ℤ) (h : a + b = n) (b' n' : ℤ)
    (hb' : a + b' = n') (hn' : n' = n + y) :
    K.ιTotal (up ℤ) a b' n' hb' ≫
      (CochainComplex.shiftFunctorObjXIso (K.total (up ℤ)) y n n' hn').inv ≫
        (K.totalShift₂Iso y).inv.f n =
      (a * y).negOnePow • (K.shiftFunctor₂XXIso a b y b' (by omega)).inv ≫
        ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) a b n h 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (K : HomologicalComplex₂ C (ComplexShape.up ℤ) (ComplexShape.up ℤ)) (y : ℤ) [K.HasTotal (ComplexShape.up ℤ)] (a b n : ℤ) (h : a + b = n) (b' n' : ℤ) (hb' : a + b' = n') (hn' : n' = n + y)
Goal: K.ιTotal (ComplexShape.up ℤ) a b' n' hb' ≫
    (CochainComplex.shiftFunctorObjXIso (K.total (ComplexShape.up ℤ)) y n n' hn').inv ≫ (K.totalShift₂Iso y).inv.f n =
  (a * y).negOnePow •
    (K.shiftFunctor₂XXIso a b y b' ⋯).inv ≫
      ((HomologicalComplex₂.shiftFunctor₂ C y).obj K).ιTotal (ComplexShape.up ℤ) a b n h
-- HomologicalComplex₂.totalShift₂Iso_hom_naturality
@[reassoc]
lemma totalShift₂Iso_hom_naturality [L.HasTotal (up ℤ)] :
    total.map ((shiftFunctor₂ C y).map f) (up ℤ) ≫ (L.totalShift₂Iso y).hom =
      (K.totalShift₂Iso y).hom ≫ (total.map f (up ℤ))⟦y⟧' 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] {K L : HomologicalComplex₂ C (ComplexShape.up ℤ) (ComplexShape.up ℤ)} (f : K ⟶ L) (y : ℤ) [K.HasTotal (ComplexShape.up ℤ)] [L.HasTotal (ComplexShape.up ℤ)]
Goal: HomologicalComplex₂.total.map ((HomologicalComplex₂.shiftFunctor₂ C y).map f) (ComplexShape.up ℤ) ≫
    (L.totalShift₂Iso y).hom =
  (K.totalShift₂Iso y).hom ≫
    (CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) y).map
      (HomologicalComplex₂.total.map f (ComplexShape.up ℤ))
-- HomologicalComplex₂.totalShift₁Iso_trans_totalShift₂Iso
lemma totalShift₁Iso_trans_totalShift₂Iso :
    ((shiftFunctor₂ C y).obj K).totalShift₁Iso x ≪≫
      (shiftFunctor (CochainComplex C ℤ) x).mapIso (K.totalShift₂Iso y) =
    (x * y).negOnePow • (total.mapIso ((shiftFunctor₁₂CommIso C x y).app K) (up ℤ)) ≪≫
      ((shiftFunctor₁ C x).obj K).totalShift₂Iso y ≪≫
      (shiftFunctor _ y).mapIso (K.totalShift₁Iso x) ≪≫
      (shiftFunctorComm (CochainComplex C ℤ) x y).app _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (K : HomologicalComplex₂ C (ComplexShape.up ℤ) (ComplexShape.up ℤ)) (x y : ℤ) [K.HasTotal (ComplexShape.up ℤ)]
Goal: ((HomologicalComplex₂.shiftFunctor₂ C y).obj K).totalShift₁Iso x ≪≫
    (CategoryTheory.shiftFunctor (CochainComplex C ℤ) x).mapIso (K.totalShift₂Iso y) =
  (x * y).negOnePow •
    HomologicalComplex₂.total.mapIso ((HomologicalComplex₂.shiftFunctor₁₂CommIso C x y).app K) (ComplexShape.up ℤ) ≪≫
      ((HomologicalComplex₂.shiftFunctor₁ C x).obj K).totalShift₂Iso y ≪≫
        (CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) y).mapIso (K.totalShift₁Iso x) ≪≫
          (CategoryTheory.shiftFunctorComm (CochainComplex C ℤ) x y).app (K.total (ComplexShape.up ℤ))
-- HomologicalComplex₂.totalShift₁Iso_hom_totalShift₂Iso_hom
@[reassoc]
lemma totalShift₁Iso_hom_totalShift₂Iso_hom :
    (((shiftFunctor₂ C y).obj K).totalShift₁Iso x).hom ≫ (K.totalShift₂Iso y).hom⟦x⟧' =
      (x * y).negOnePow • (total.map ((shiftFunctor₁₂CommIso C x y).hom.app K) (up ℤ) ≫
          (((shiftFunctor₁ C x).obj K).totalShift₂Iso y).hom ≫
          (K.totalShift₁Iso x).hom⟦y⟧' ≫
          (shiftFunctorComm (CochainComplex C ℤ) x y).hom.app _) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.Preadditive C] (K : HomologicalComplex₂ C (ComplexShape.up ℤ) (ComplexShape.up ℤ)) (x y : ℤ) [K.HasTotal (ComplexShape.up ℤ)]
Goal: (((HomologicalComplex₂.shiftFunctor₂ C y).obj K).totalShift₁Iso x).hom ≫
    (CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) x).map (K.totalShift₂Iso y).hom =
  (x * y).negOnePow •
    HomologicalComplex₂.total.map ((HomologicalComplex₂.shiftFunctor₁₂CommIso C x y).hom.app K) (ComplexShape.up ℤ) ≫
      (((HomologicalComplex₂.shiftFunctor₁ C x).obj K).totalShift₂Iso y).hom ≫
        (CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) y).map (K.totalShift₁Iso x).hom ≫
          (CategoryTheory.shiftFunctorComm (CochainComplex C ℤ) x y).hom.app (K.total (ComplexShape.up ℤ))
-- ChainComplex.truncate_obj_d
@[simps]
def truncate [HasZeroMorphisms V] : ChainComplex V ℕ ⥤ ChainComplex V ℕ 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) (i j : ℕ)
Goal: (ChainComplex.truncate.obj C).d i j = C.d (i + 1) (j + 1)
-- ChainComplex.truncate_obj_X
@[simps]
def truncate [HasZeroMorphisms V] : ChainComplex V ℕ ⥤ ChainComplex V ℕ 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) (i : ℕ)
Goal: (ChainComplex.truncate.obj C).X i = C.X (i + 1)
-- ChainComplex.truncate_map_f
@[simps]
def truncate [HasZeroMorphisms V] : ChainComplex V ℕ ⥤ ChainComplex V ℕ 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V]
Goal: ∀ {X Y : ChainComplex V ℕ} (f : X ⟶ Y) (i : ℕ), (ChainComplex.truncate.map f).f i = f.f (i + 1)
-- ChainComplex.augment_X_zero
@[simp]
theorem augment_X_zero (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0) :
    (augment C f w).X 0 = X 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0)
Goal: (C.augment f w).X 0 = X
-- ChainComplex.augment_X_succ
@[simp]
theorem augment_X_succ (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0)
    (i : ℕ) : (augment C f w).X (i + 1) = C.X i 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0) (i : ℕ)
Goal: (C.augment f w).X (i + 1) = C.X i
-- ChainComplex.augment_d_one_zero
@[simp]
theorem augment_d_one_zero (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0) :
    (augment C f w).d 1 0 = f 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0)
Goal: (C.augment f w).d 1 0 = f
-- ChainComplex.augment_d_succ_succ
@[simp]
theorem augment_d_succ_succ (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0)
    (i j : ℕ) : (augment C f w).d (i + 1) (j + 1) = C.d i j 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0) (i j : ℕ)
Goal: (C.augment f w).d (i + 1) (j + 1) = C.d i j
-- ChainComplex.truncateAugment_hom_f
@[simp]
theorem truncateAugment_hom_f (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0)
    (i : ℕ) : (truncateAugment C f w).hom.f i = 𝟙 (C.X i) 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0) (i : ℕ)
Goal: (C.truncateAugment f w).hom.f i = 𝟙 (C.X i)
-- ChainComplex.truncateAugment_inv_f
@[simp]
theorem truncateAugment_inv_f (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0)
    (i : ℕ) : (truncateAugment C f w).inv.f i = 𝟙 ((truncate.obj (augment C f w)).X i) 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0) (i : ℕ)
Goal: (C.truncateAugment f w).inv.f i = 𝟙 ((ChainComplex.truncate.obj (C.augment f w)).X i)
-- ChainComplex.chainComplex_d_succ_succ_zero
@[simp]
theorem chainComplex_d_succ_succ_zero (C : ChainComplex V ℕ) (i : ℕ) : C.d (i + 2) 0 = 0 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) (i : ℕ)
Goal: C.d (i + 2) 0 = 0
-- ChainComplex.augmentTruncate_hom_f_zero
@[simp]
theorem augmentTruncate_hom_f_zero (C : ChainComplex V ℕ) :
    (augmentTruncate C).hom.f 0 = 𝟙 (C.X 0) 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ)
Goal: C.augmentTruncate.hom.f 0 = 𝟙 (C.X 0)
-- ChainComplex.augmentTruncate_hom_f_succ
@[simp]
theorem augmentTruncate_hom_f_succ (C : ChainComplex V ℕ) (i : ℕ) :
    (augmentTruncate C).hom.f (i + 1) = 𝟙 (C.X (i + 1)) 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) (i : ℕ)
Goal: C.augmentTruncate.hom.f (i + 1) = 𝟙 (C.X (i + 1))
-- ChainComplex.augmentTruncate_inv_f_zero
@[simp]
theorem augmentTruncate_inv_f_zero (C : ChainComplex V ℕ) :
    (augmentTruncate C).inv.f 0 = 𝟙 (C.X 0) 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ)
Goal: C.augmentTruncate.inv.f 0 = 𝟙 (C.X 0)
-- ChainComplex.augmentTruncate_inv_f_succ
@[simp]
theorem augmentTruncate_inv_f_succ (C : ChainComplex V ℕ) (i : ℕ) :
    (augmentTruncate C).inv.f (i + 1) = 𝟙 (C.X (i + 1)) 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : ChainComplex V ℕ) (i : ℕ)
Goal: C.augmentTruncate.inv.f (i + 1) = 𝟙 (C.X (i + 1))
-- CochainComplex.truncate_obj_X
@[simps]
def truncate [HasZeroMorphisms V] : CochainComplex V ℕ ⥤ CochainComplex V ℕ 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) (i : ℕ)
Goal: (CochainComplex.truncate.obj C).X i = C.X (i + 1)
-- CochainComplex.truncate_obj_d
@[simps]
def truncate [HasZeroMorphisms V] : CochainComplex V ℕ ⥤ CochainComplex V ℕ 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) (i j : ℕ)
Goal: (CochainComplex.truncate.obj C).d i j = C.d (i + 1) (j + 1)
-- CochainComplex.truncate_map_f
@[simps]
def truncate [HasZeroMorphisms V] : CochainComplex V ℕ ⥤ CochainComplex V ℕ 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V]
Goal: ∀ {X Y : CochainComplex V ℕ} (f : X ⟶ Y) (i : ℕ), (CochainComplex.truncate.map f).f i = f.f (i + 1)
-- CochainComplex.augment_X_zero
@[simp]
theorem augment_X_zero (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0) :
    (augment C f w).X 0 = X 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0)
Goal: (C.augment f w).X 0 = X
-- CochainComplex.augment_X_succ
@[simp]
theorem augment_X_succ (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0)
    (i : ℕ) : (augment C f w).X (i + 1) = C.X i 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0) (i : ℕ)
Goal: (C.augment f w).X (i + 1) = C.X i
-- CochainComplex.augment_d_zero_one
@[simp]
theorem augment_d_zero_one (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0) :
    (augment C f w).d 0 1 = f 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0)
Goal: (C.augment f w).d 0 1 = f
-- CochainComplex.augment_d_succ_succ
@[simp]
theorem augment_d_succ_succ (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0)
    (i j : ℕ) : (augment C f w).d (i + 1) (j + 1) = C.d i j 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0) (i j : ℕ)
Goal: (C.augment f w).d (i + 1) (j + 1) = C.d i j
-- CochainComplex.truncateAugment_hom_f
@[simp]
theorem truncateAugment_hom_f (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0)
    (w : f ≫ C.d 0 1 = 0) (i : ℕ) : (truncateAugment C f w).hom.f i = 𝟙 (C.X i) 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0) (i : ℕ)
Goal: (C.truncateAugment f w).hom.f i = 𝟙 (C.X i)
-- CochainComplex.truncateAugment_inv_f
@[simp]
theorem truncateAugment_inv_f (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0)
    (w : f ≫ C.d 0 1 = 0) (i : ℕ) :
    (truncateAugment C f w).inv.f i = 𝟙 ((truncate.obj (augment C f w)).X i) 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) {X : V} (f : X ⟶ C.X 0) (w : f ≫ C.d 0 1 = 0) (i : ℕ)
Goal: (C.truncateAugment f w).inv.f i = 𝟙 ((CochainComplex.truncate.obj (C.augment f w)).X i)
-- CochainComplex.cochainComplex_d_succ_succ_zero
@[simp]
theorem cochainComplex_d_succ_succ_zero (C : CochainComplex V ℕ) (i : ℕ) : C.d 0 (i + 2) = 0 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ) (i : ℕ)
Goal: C.d 0 (i + 2) = 0
-- CochainComplex.augmentTruncate_hom_f_zero
@[simp]
theorem augmentTruncate_hom_f_zero (C : CochainComplex V ℕ) :
    (augmentTruncate C).hom.f 0 = 𝟙 (C.X 0) 
Arguments: {V : Type u} [CategoryTheory.Category.{v, u} V] [CategoryTheory.Limits.HasZeroMorphisms V] (C : CochainComplex V ℕ)
Goal: C.augmentTruncate.hom.f 0 = 𝟙 (C.X 0)