ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CochainComplex.mappingCone.inl_descCochain
@[simp]
lemma inl_descCochain :
    (inl φ).comp (descCochain φ α β h) (by omega) = α 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} {n m : ℤ} (α : CochainComplex.HomComplex.Cochain F K m) (β : CochainComplex.HomComplex.Cochain G K n) (h : m + 1 = n)
Goal: (CochainComplex.mappingCone.inl φ).comp (CochainComplex.mappingCone.descCochain φ α β h) ⋯ = α
-- CochainComplex.mappingCone.inr_descCochain
@[simp]
lemma inr_descCochain :
    (Cochain.ofHom (inr φ)).comp (descCochain φ α β h) (zero_add n) = β 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} {n m : ℤ} (α : CochainComplex.HomComplex.Cochain F K m) (β : CochainComplex.HomComplex.Cochain G K n) (h : m + 1 = n)
Goal: (CochainComplex.HomComplex.Cochain.ofHom (CochainComplex.mappingCone.inr φ)).comp
    (CochainComplex.mappingCone.descCochain φ α β h) ⋯ =
  β
-- CochainComplex.mappingCone.inl_v_descCochain_v
@[reassoc (attr := simp)]
lemma inl_v_descCochain_v (p₁ p₂ p₃ : ℤ) (h₁₂ : p₁ + (-1) = p₂) (h₂₃ : p₂ + n = p₃) :
    (inl φ).v p₁ p₂ h₁₂ ≫ (descCochain φ α β h).v p₂ p₃ h₂₃ =
        α.v p₁ p₃ (by rw [← h₂₃, ← h₁₂, ← h, add_comm m, add_assoc, neg_add_cancel_left]) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} {n m : ℤ} (α : CochainComplex.HomComplex.Cochain F K m) (β : CochainComplex.HomComplex.Cochain G K n) (h : m + 1 = n) (p₁ p₂ p₃ : ℤ) (h₁₂ : p₁ + -1 = p₂) (h₂₃ : p₂ + n = p₃)
Goal: (CochainComplex.mappingCone.inl φ).v p₁ p₂ h₁₂ ≫ (CochainComplex.mappingCone.descCochain φ α β h).v p₂ p₃ h₂₃ =
  α.v p₁ p₃ ⋯
-- CochainComplex.mappingCone.inr_f_descCochain_v
@[reassoc (attr := simp)]
lemma inr_f_descCochain_v (p₁ p₂ : ℤ) (h₁₂ : p₁ + n = p₂) :
    (inr φ).f p₁ ≫ (descCochain φ α β h).v p₁ p₂ h₁₂ = β.v p₁ p₂ h₁₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} {n m : ℤ} (α : CochainComplex.HomComplex.Cochain F K m) (β : CochainComplex.HomComplex.Cochain G K n) (h : m + 1 = n) (p₁ p₂ : ℤ) (h₁₂ : p₁ + n = p₂)
Goal: (CochainComplex.mappingCone.inr φ).f p₁ ≫ (CochainComplex.mappingCone.descCochain φ α β h).v p₁ p₂ h₁₂ = β.v p₁ p₂ h₁₂
-- CochainComplex.mappingCone.δ_descCochain
lemma δ_descCochain (n' : ℤ) (hn' : n + 1 = n') :
    δ n n' (descCochain φ α β h) =
      (fst φ).1.comp (δ m n α +
          n'.negOnePow • (Cochain.ofHom φ).comp β (zero_add n)) (by omega) +
      (snd φ).comp (δ n n' β) (zero_add n') 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} {n m : ℤ} (α : CochainComplex.HomComplex.Cochain F K m) (β : CochainComplex.HomComplex.Cochain G K n) (h : m + 1 = n) (n' : ℤ) (hn' : n + 1 = n')
Goal: CochainComplex.HomComplex.δ n n' (CochainComplex.mappingCone.descCochain φ α β h) =
  (↑(CochainComplex.mappingCone.fst φ)).comp
      (CochainComplex.HomComplex.δ m n α + n'.negOnePow • (CochainComplex.HomComplex.Cochain.ofHom φ).comp β ⋯) ⋯ +
    (CochainComplex.mappingCone.snd φ).comp (CochainComplex.HomComplex.δ n n' β) ⋯
-- CochainComplex.mappingCone.descCocycle_coe
@[simps!]
noncomputable def descCocycle {K : CochainComplex C ℤ} {n m : ℤ}
    (α : Cochain F K m) (β : Cocycle G K n)
    (h : m + 1 = n) (eq : δ m n α = n.negOnePow • (Cochain.ofHom φ).comp β.1 (zero_add n)) :
    Cocycle (mappingCone φ) K n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} {n m : ℤ} (α : CochainComplex.HomComplex.Cochain F K m) (β : CochainComplex.HomComplex.Cocycle G K n) (h : m + 1 = n) (eq : CochainComplex.HomComplex.δ m n α = n.negOnePow • (CochainComplex.HomComplex.Cochain.ofHom φ).comp ↑β ⋯)
Goal: ↑(CochainComplex.mappingCone.descCocycle φ α β h eq) = CochainComplex.mappingCone.descCochain φ α (↑β) h
-- CochainComplex.mappingCone.ofHom_desc
@[simp]
lemma ofHom_desc :
    Cochain.ofHom (desc φ α β eq) = descCochain φ α (Cochain.ofHom β) (neg_add_self 1) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} (α : CochainComplex.HomComplex.Cochain F K (-1)) (β : G ⟶ K) (eq : CochainComplex.HomComplex.δ (-1) 0 α = CochainComplex.HomComplex.Cochain.ofHom (φ ≫ β))
Goal: CochainComplex.HomComplex.Cochain.ofHom (CochainComplex.mappingCone.desc φ α β eq) =
  CochainComplex.mappingCone.descCochain φ α (CochainComplex.HomComplex.Cochain.ofHom β) ⋯
-- CochainComplex.mappingCone.inl_v_desc_f
@[reassoc (attr := simp)]
lemma inl_v_desc_f (p q : ℤ) (h : p + (-1) = q) :
    (inl φ).v p q h ≫ (desc φ α β eq).f q = α.v p q h 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} (α : CochainComplex.HomComplex.Cochain F K (-1)) (β : G ⟶ K) (eq : CochainComplex.HomComplex.δ (-1) 0 α = CochainComplex.HomComplex.Cochain.ofHom (φ ≫ β)) (p q : ℤ) (h : p + -1 = q)
Goal: (CochainComplex.mappingCone.inl φ).v p q h ≫ (CochainComplex.mappingCone.desc φ α β eq).f q = α.v p q h
-- CochainComplex.mappingCone.inl_desc
lemma inl_desc :
    (inl φ).comp (Cochain.ofHom (desc φ α β eq)) (add_zero _) = α 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} (α : CochainComplex.HomComplex.Cochain F K (-1)) (β : G ⟶ K) (eq : CochainComplex.HomComplex.δ (-1) 0 α = CochainComplex.HomComplex.Cochain.ofHom (φ ≫ β))
Goal: (CochainComplex.mappingCone.inl φ).comp
    (CochainComplex.HomComplex.Cochain.ofHom (CochainComplex.mappingCone.desc φ α β eq)) ⋯ =
  α
-- CochainComplex.mappingCone.inr_f_desc_f
@[reassoc (attr := simp)]
lemma inr_f_desc_f (p : ℤ) :
    (inr φ).f p ≫ (desc φ α β eq).f p = β.f p 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} (α : CochainComplex.HomComplex.Cochain F K (-1)) (β : G ⟶ K) (eq : CochainComplex.HomComplex.δ (-1) 0 α = CochainComplex.HomComplex.Cochain.ofHom (φ ≫ β)) (p : ℤ)
Goal: (CochainComplex.mappingCone.inr φ).f p ≫ (CochainComplex.mappingCone.desc φ α β eq).f p = β.f p
-- CochainComplex.mappingCone.inr_desc
@[reassoc (attr := simp)]
lemma inr_desc : inr φ ≫ desc φ α β eq = β 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} (α : CochainComplex.HomComplex.Cochain F K (-1)) (β : G ⟶ K) (eq : CochainComplex.HomComplex.δ (-1) 0 α = CochainComplex.HomComplex.Cochain.ofHom (φ ≫ β))
Goal: CochainComplex.mappingCone.inr φ ≫ CochainComplex.mappingCone.desc φ α β eq = β
-- CochainComplex.mappingCone.desc_f
lemma desc_f (p q : ℤ) (hpq : p + 1 = q) :
    (desc φ α β eq).f p = (fst φ).1.v p q hpq ≫ α.v q p (by omega) +
      (snd φ).v p p (add_zero p) ≫ β.f p 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} (α : CochainComplex.HomComplex.Cochain F K (-1)) (β : G ⟶ K) (eq : CochainComplex.HomComplex.δ (-1) 0 α = CochainComplex.HomComplex.Cochain.ofHom (φ ≫ β)) (p q : ℤ) (hpq : p + 1 = q)
Goal: (CochainComplex.mappingCone.desc φ α β eq).f p =
  (↑(CochainComplex.mappingCone.fst φ)).v p q hpq ≫ α.v q p ⋯ + (CochainComplex.mappingCone.snd φ).v p p ⋯ ≫ β.f p
-- CochainComplex.mappingCone.liftCochain_fst
@[simp]
lemma liftCochain_fst :
    (liftCochain φ α β h).comp (fst φ).1 h = α 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} {n m : ℤ} (α : CochainComplex.HomComplex.Cochain K F m) (β : CochainComplex.HomComplex.Cochain K G n) (h : n + 1 = m)
Goal: (CochainComplex.mappingCone.liftCochain φ α β h).comp (↑(CochainComplex.mappingCone.fst φ)) h = α
-- CochainComplex.mappingCone.liftCochain_snd
@[simp]
lemma liftCochain_snd :
    (liftCochain φ α β h).comp (snd φ) (add_zero n) = β 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} {n m : ℤ} (α : CochainComplex.HomComplex.Cochain K F m) (β : CochainComplex.HomComplex.Cochain K G n) (h : n + 1 = m)
Goal: (CochainComplex.mappingCone.liftCochain φ α β h).comp (CochainComplex.mappingCone.snd φ) ⋯ = β
-- CochainComplex.mappingCone.liftCochain_v_fst_v
@[reassoc (attr := simp)]
lemma liftCochain_v_fst_v (p₁ p₂ p₃ : ℤ) (h₁₂ : p₁ + n = p₂) (h₂₃ : p₂ + 1 = p₃) :
    (liftCochain φ α β h).v p₁ p₂ h₁₂ ≫ (fst φ).1.v p₂ p₃ h₂₃ = α.v p₁ p₃ (by omega) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} {n m : ℤ} (α : CochainComplex.HomComplex.Cochain K F m) (β : CochainComplex.HomComplex.Cochain K G n) (h : n + 1 = m) (p₁ p₂ p₃ : ℤ) (h₁₂ : p₁ + n = p₂) (h₂₃ : p₂ + 1 = p₃)
Goal: (CochainComplex.mappingCone.liftCochain φ α β h).v p₁ p₂ h₁₂ ≫ (↑(CochainComplex.mappingCone.fst φ)).v p₂ p₃ h₂₃ =
  α.v p₁ p₃ ⋯
-- CochainComplex.mappingCone.liftCochain_v_snd_v
@[reassoc (attr := simp)]
lemma liftCochain_v_snd_v (p₁ p₂ : ℤ) (h₁₂ : p₁ + n = p₂) :
    (liftCochain φ α β h).v p₁ p₂ h₁₂ ≫ (snd φ).v p₂ p₂ (add_zero p₂) = β.v p₁ p₂ h₁₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} {n m : ℤ} (α : CochainComplex.HomComplex.Cochain K F m) (β : CochainComplex.HomComplex.Cochain K G n) (h : n + 1 = m) (p₁ p₂ : ℤ) (h₁₂ : p₁ + n = p₂)
Goal: (CochainComplex.mappingCone.liftCochain φ α β h).v p₁ p₂ h₁₂ ≫ (CochainComplex.mappingCone.snd φ).v p₂ p₂ ⋯ =
  β.v p₁ p₂ h₁₂
-- CochainComplex.mappingCone.δ_liftCochain
lemma δ_liftCochain (m' : ℤ) (hm' : m + 1 = m') :
    δ n m (liftCochain φ α β h) = -(δ m m' α).comp (inl φ) (by omega) +
      (δ n m β + α.comp (Cochain.ofHom φ) (add_zero m)).comp
        (Cochain.ofHom (inr φ)) (add_zero m) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} {n m : ℤ} (α : CochainComplex.HomComplex.Cochain K F m) (β : CochainComplex.HomComplex.Cochain K G n) (h : n + 1 = m) (m' : ℤ) (hm' : m + 1 = m')
Goal: CochainComplex.HomComplex.δ n m (CochainComplex.mappingCone.liftCochain φ α β h) =
  -(CochainComplex.HomComplex.δ m m' α).comp (CochainComplex.mappingCone.inl φ) ⋯ +
    (CochainComplex.HomComplex.δ n m β + α.comp (CochainComplex.HomComplex.Cochain.ofHom φ) ⋯).comp
      (CochainComplex.HomComplex.Cochain.ofHom (CochainComplex.mappingCone.inr φ)) ⋯
-- CochainComplex.mappingCone.liftCocycle_coe
@[simps!]
noncomputable def liftCocycle {K : CochainComplex C ℤ} {n m : ℤ}
    (α : Cocycle K F m) (β : Cochain K G n) (h : n + 1 = m)
    (eq : δ n m β + α.1.comp (Cochain.ofHom φ) (add_zero m) = 0) :
    Cocycle K (mappingCone φ) n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} {n m : ℤ} (α : CochainComplex.HomComplex.Cocycle K F m) (β : CochainComplex.HomComplex.Cochain K G n) (h : n + 1 = m) (eq : CochainComplex.HomComplex.δ n m β + (↑α).comp (CochainComplex.HomComplex.Cochain.ofHom φ) ⋯ = 0)
Goal: ↑(CochainComplex.mappingCone.liftCocycle φ α β h eq) = CochainComplex.mappingCone.liftCochain φ (↑α) β h
-- CochainComplex.mappingCone.ofHom_lift
@[simp]
lemma ofHom_lift :
    Cochain.ofHom (lift φ α β eq) = liftCochain φ α β (zero_add 1) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} (α : CochainComplex.HomComplex.Cocycle K F 1) (β : CochainComplex.HomComplex.Cochain K G 0) (eq : CochainComplex.HomComplex.δ 0 1 β + (↑α).comp (CochainComplex.HomComplex.Cochain.ofHom φ) ⋯ = 0)
Goal: CochainComplex.HomComplex.Cochain.ofHom (CochainComplex.mappingCone.lift φ α β eq) =
  CochainComplex.mappingCone.liftCochain φ (↑α) β ⋯
-- CochainComplex.mappingCone.lift_f_fst_v
@[reassoc (attr := simp)]
lemma lift_f_fst_v (p q : ℤ) (hpq : p + 1 = q) :
    (lift φ α β eq).f p ≫ (fst φ).1.v p q hpq = α.1.v p q hpq 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} (α : CochainComplex.HomComplex.Cocycle K F 1) (β : CochainComplex.HomComplex.Cochain K G 0) (eq : CochainComplex.HomComplex.δ 0 1 β + (↑α).comp (CochainComplex.HomComplex.Cochain.ofHom φ) ⋯ = 0) (p q : ℤ) (hpq : p + 1 = q)
Goal: (CochainComplex.mappingCone.lift φ α β eq).f p ≫ (↑(CochainComplex.mappingCone.fst φ)).v p q hpq = (↑α).v p q hpq
-- CochainComplex.mappingCone.lift_fst
lemma lift_fst :
    (Cochain.ofHom (lift φ α β eq)).comp (fst φ).1 (zero_add 1) = α.1 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} (α : CochainComplex.HomComplex.Cocycle K F 1) (β : CochainComplex.HomComplex.Cochain K G 0) (eq : CochainComplex.HomComplex.δ 0 1 β + (↑α).comp (CochainComplex.HomComplex.Cochain.ofHom φ) ⋯ = 0)
Goal: (CochainComplex.HomComplex.Cochain.ofHom (CochainComplex.mappingCone.lift φ α β eq)).comp
    ↑(CochainComplex.mappingCone.fst φ) ⋯ =
  ↑α
-- CochainComplex.mappingCone.lift_f_snd_v
@[reassoc (attr := simp)]
lemma lift_f_snd_v (p q : ℤ) (hpq : p + 0 = q) :
    (lift φ α β eq).f p ≫ (snd φ).v p q hpq = β.v p q hpq 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} (α : CochainComplex.HomComplex.Cocycle K F 1) (β : CochainComplex.HomComplex.Cochain K G 0) (eq : CochainComplex.HomComplex.δ 0 1 β + (↑α).comp (CochainComplex.HomComplex.Cochain.ofHom φ) ⋯ = 0) (p q : ℤ) (hpq : p + 0 = q)
Goal: (CochainComplex.mappingCone.lift φ α β eq).f p ≫ (CochainComplex.mappingCone.snd φ).v p q hpq = β.v p q hpq
-- CochainComplex.mappingCone.lift_snd
lemma lift_snd :
    (Cochain.ofHom (lift φ α β eq)).comp (snd φ) (zero_add 0) = β 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} (α : CochainComplex.HomComplex.Cocycle K F 1) (β : CochainComplex.HomComplex.Cochain K G 0) (eq : CochainComplex.HomComplex.δ 0 1 β + (↑α).comp (CochainComplex.HomComplex.Cochain.ofHom φ) ⋯ = 0)
Goal: (CochainComplex.HomComplex.Cochain.ofHom (CochainComplex.mappingCone.lift φ α β eq)).comp
    (CochainComplex.mappingCone.snd φ) ⋯ =
  β
-- CochainComplex.mappingCone.lift_f
lemma lift_f (p q : ℤ) (hpq : p + 1 = q) :
    (lift φ α β eq).f p = α.1.v p q hpq ≫
      (inl φ).v q p (by omega) + β.v p p (add_zero p) ≫ (inr φ).f p 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K : CochainComplex C ℤ} (α : CochainComplex.HomComplex.Cocycle K F 1) (β : CochainComplex.HomComplex.Cochain K G 0) (eq : CochainComplex.HomComplex.δ 0 1 β + (↑α).comp (CochainComplex.HomComplex.Cochain.ofHom φ) ⋯ = 0) (p q : ℤ) (hpq : p + 1 = q)
Goal: (CochainComplex.mappingCone.lift φ α β eq).f p =
  (↑α).v p q hpq ≫ (CochainComplex.mappingCone.inl φ).v q p ⋯ + β.v p p ⋯ ≫ (CochainComplex.mappingCone.inr φ).f p
-- CochainComplex.mappingCone.liftCochain_descCochain
@[simp]
lemma liftCochain_descCochain :
    (liftCochain φ α β h).comp (descCochain φ α' β' h') hp =
      α.comp α' (by omega) + β.comp β' (by omega) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K L : CochainComplex C ℤ} {n m : ℤ} (α : CochainComplex.HomComplex.Cochain K F m) (β : CochainComplex.HomComplex.Cochain K G n) {n' m' : ℤ} (α' : CochainComplex.HomComplex.Cochain F L m') (β' : CochainComplex.HomComplex.Cochain G L n') (h : n + 1 = m) (h' : m' + 1 = n') (p : ℤ) (hp : n + n' = p)
Goal: (CochainComplex.mappingCone.liftCochain φ α β h).comp (CochainComplex.mappingCone.descCochain φ α' β' h') hp =
  α.comp α' ⋯ + β.comp β' ⋯
-- CochainComplex.mappingCone.liftCochain_v_descCochain_v
lemma liftCochain_v_descCochain_v (p₁ p₂ p₃ : ℤ) (h₁₂ : p₁ + n = p₂) (h₂₃ : p₂ + n' = p₃)
    (q : ℤ) (hq : p₁ + m = q) :
    (liftCochain φ α β h).v p₁ p₂ h₁₂ ≫ (descCochain φ α' β' h').v p₂ p₃ h₂₃ =
      α.v p₁ q hq ≫ α'.v q p₃ (by omega) + β.v p₁ p₂ h₁₂ ≫ β'.v p₂ p₃ h₂₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K L : CochainComplex C ℤ} {n m : ℤ} (α : CochainComplex.HomComplex.Cochain K F m) (β : CochainComplex.HomComplex.Cochain K G n) {n' m' : ℤ} (α' : CochainComplex.HomComplex.Cochain F L m') (β' : CochainComplex.HomComplex.Cochain G L n') (h : n + 1 = m) (h' : m' + 1 = n') (p : ℤ) (hp : n + n' = p) (p₁ p₂ p₃ : ℤ) (h₁₂ : p₁ + n = p₂) (h₂₃ : p₂ + n' = p₃) (q : ℤ) (hq : p₁ + m = q)
Goal: (CochainComplex.mappingCone.liftCochain φ α β h).v p₁ p₂ h₁₂ ≫
    (CochainComplex.mappingCone.descCochain φ α' β' h').v p₂ p₃ h₂₃ =
  α.v p₁ q hq ≫ α'.v q p₃ ⋯ + β.v p₁ p₂ h₁₂ ≫ β'.v p₂ p₃ h₂₃
-- CochainComplex.mappingCone.lift_desc_f
lemma lift_desc_f {K L : CochainComplex C ℤ} (α : Cocycle K F 1) (β : Cochain K G 0)
    (eq : δ 0 1 β + α.1.comp (Cochain.ofHom φ) (add_zero 1) = 0)
    (α' : Cochain F L (-1)) (β' : G ⟶ L)
    (eq' : δ (-1) 0 α' = Cochain.ofHom (φ ≫ β')) (n n' : ℤ) (hnn' : n + 1 = n') :
    (lift φ α β eq).f n ≫ (desc φ α' β' eq').f n =
    α.1.v n n' hnn' ≫ α'.v n' n (by omega) + β.v n n (add_zero n) ≫ β'.f n 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Preadditive C] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] {K L : CochainComplex C ℤ} (α : CochainComplex.HomComplex.Cocycle K F 1) (β : CochainComplex.HomComplex.Cochain K G 0) (eq : CochainComplex.HomComplex.δ 0 1 β + (↑α).comp (CochainComplex.HomComplex.Cochain.ofHom φ) ⋯ = 0) (α' : CochainComplex.HomComplex.Cochain F L (-1)) (β' : G ⟶ L) (eq' : CochainComplex.HomComplex.δ (-1) 0 α' = CochainComplex.HomComplex.Cochain.ofHom (φ ≫ β')) (n n' : ℤ) (hnn' : n + 1 = n')
Goal: (CochainComplex.mappingCone.lift φ α β eq).f n ≫ (CochainComplex.mappingCone.desc φ α' β' eq').f n =
  (↑α).v n n' hnn' ≫ α'.v n' n ⋯ + β.v n n ⋯ ≫ β'.f n
-- CochainComplex.mappingCone.mapHomologicalComplexXIso'_hom
@[simps]
noncomputable def mapHomologicalComplexXIso' (n m : ℤ) (hnm : n + 1 = m) :
    ((H.mapHomologicalComplex (ComplexShape.up ℤ)).obj (mappingCone φ)).X n ≅
      (mappingCone ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).X n 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] (H : C ⥤ D) [H.Additive] [HomologicalComplex.HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)] (n m : ℤ) (hnm : n + 1 = m)
Goal: (CochainComplex.mappingCone.mapHomologicalComplexXIso' φ H n m hnm).hom =
  H.map ((↑(CochainComplex.mappingCone.fst φ)).v n m ⋯) ≫
      (CochainComplex.mappingCone.inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v m n ⋯ +
    H.map ((CochainComplex.mappingCone.snd φ).v n n ⋯) ≫
      (CochainComplex.mappingCone.inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n
-- CochainComplex.mappingCone.mapHomologicalComplexXIso'_inv
@[simps]
noncomputable def mapHomologicalComplexXIso' (n m : ℤ) (hnm : n + 1 = m) :
    ((H.mapHomologicalComplex (ComplexShape.up ℤ)).obj (mappingCone φ)).X n ≅
      (mappingCone ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).X n 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] (H : C ⥤ D) [H.Additive] [HomologicalComplex.HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)] (n m : ℤ) (hnm : n + 1 = m)
Goal: (CochainComplex.mappingCone.mapHomologicalComplexXIso' φ H n m hnm).inv =
  (↑(CochainComplex.mappingCone.fst ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))).v n m ⋯ ≫
      H.map ((CochainComplex.mappingCone.inl φ).v m n ⋯) +
    (CochainComplex.mappingCone.snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v n n ⋯ ≫
      H.map ((CochainComplex.mappingCone.inr φ).f n)
-- CochainComplex.mappingCone.mapHomologicalComplexXIso_eq
lemma mapHomologicalComplexXIso_eq (n m : ℤ) (hnm : n + 1 = m) :
    mapHomologicalComplexXIso φ H n = mapHomologicalComplexXIso' φ H n m hnm 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_3, u_2} D] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] (H : C ⥤ D) [H.Additive] [HomologicalComplex.HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)] (n m : ℤ) (hnm : n + 1 = m)
Goal: CochainComplex.mappingCone.mapHomologicalComplexXIso φ H n =
  CochainComplex.mappingCone.mapHomologicalComplexXIso' φ H n m hnm
-- CochainComplex.mappingCone.map_inr
lemma map_inr :
    (H.mapHomologicalComplex (ComplexShape.up ℤ)).map (inr φ) ≫
      (mapHomologicalComplexIso φ H).hom =
    inr ((Functor.mapHomologicalComplex H (ComplexShape.up ℤ)).map φ) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_3, u_2} D] [CategoryTheory.Preadditive C] [CategoryTheory.Preadditive D] {F G : CochainComplex C ℤ} (φ : F ⟶ G) [HomologicalComplex.HasHomotopyCofiber φ] (H : C ⥤ D) [H.Additive] [HomologicalComplex.HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)]
Goal: (H.mapHomologicalComplex (ComplexShape.up ℤ)).map (CochainComplex.mappingCone.inr φ) ≫
    (CochainComplex.mappingCone.mapHomologicalComplexIso φ H).hom =
  CochainComplex.mappingCone.inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)
-- HomologicalComplex.extend.isZero_X
lemma isZero_X {i : Option ι} (hi : i = none) :
    IsZero (X K i) 
Arguments: {ι : Type u_1} {c : ComplexShape ι} {C : Type u_3} [CategoryTheory.Category.{u_4, u_3} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroMorphisms C] (K : HomologicalComplex C c) {i : Option ι} (hi : i = none)
Goal: CategoryTheory.Limits.IsZero (HomologicalComplex.extend.X K i)