awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
n : ℕ
R₁ R₂ : ComposableArrows.{v_1, u_1} C n
hR₁ : R₁.Exact
hR₂ : R₂.Exact
φ : R₁ ⟶ R₂
k₀ : ℕ
h : k₀ + 4 ≤ n
h₁ :
  IsIso (X := R₁.obj' (k₀ + 1) ⋯) (Y := R₂.obj' (k₀ + 1) ⋯)
    (app' (Functor.map (natAddLEFunctor h) φ) 1 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_4 :
      (Functor.obj (natAddLEFunctor h) R₁).obj' 1 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_4 ⟶
        (Functor.obj (natAddLEFunctor h) R₂).obj' 1 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_4)
h₃ :
  IsIso (X := R₁.obj' (k₀ + 3) ⋯) (Y := R₂.obj' (k₀ + 3) ⋯)
    (app' (Functor.map (natAddLEFunctor h) φ) 3 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_6 :
      (Functor.obj (natAddLEFunctor h) R₁).obj' 3 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_6 ⟶
        (Functor.obj (natAddLEFunctor h) R₂).obj' 3 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_6)
h₄ :
  Mono (X := R₁.obj' (k₀ + 4) ⋯) (Y := R₂.obj' (k₀ + 4) ⋯)
    (app' (Functor.map (natAddLEFunctor h) φ) 4 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_7 :
      (Functor.obj (natAddLEFunctor h) R₁).obj' 4 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_7 ⟶
        (Functor.obj (natAddLEFunctor h) R₂).obj' 4 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_7)
h₀ :
  Epi (X := R₁.obj' (k₀ + 0) ⋯) (Y := R₂.obj' (k₀ + 0) ⋯)
    (app' (Functor.map (natAddLEFunctor h) φ) 0 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_2 :
      (Functor.obj (natAddLEFunctor h) R₁).obj' 0 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_2 ⟶
        (Functor.obj (natAddLEFunctor h) R₂).obj' 0 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_2)
⊢ IsIso (X := R₁.obj' (k₀ + 2) ⋯) (Y := R₂.obj' (k₀ + 2) ⋯)
    (app' (Functor.map (natAddLEFunctor h) φ) 2 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_9 :
      (Functor.obj (natAddLEFunctor h) R₁).obj' 2 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_9 ⟶
        (Functor.obj (natAddLEFunctor h) R₂).obj' 2 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_9)

Tactic:
isIso_of_epi_of_isIso_of_isIso_of_mono (natAddLEFunctor_obj_exact h hR₁)
    (natAddLEFunctor_obj_exact h hR₂) _ h₀ h₁ h₃ h₄

Premises used: CategoryTheory.Abelian.isIso_of_epi_of_isIso_of_isIso_of_mono, CategoryTheory.ComposableArrows.natAddLEFunctor_obj_exact
-- Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
n : ℕ
R₁ R₂ : ComposableArrows.{v_1, u_1} C n
hR₁ : R₁.Exact
hR₂ : R₂.Exact
φ : R₁ ⟶ R₂
k₀ : ℕ
h : k₀ + 4 ≤ n
h₁ :
  IsIso (X := R₁.obj' (k₀ + 1) ⋯) (Y := R₂.obj' (k₀ + 1) ⋯)
    (app' (Functor.map (natAddLEFunctor h) φ) 1 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_4 :
      (Functor.obj (natAddLEFunctor h) R₁).obj' 1 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_4 ⟶
        (Functor.obj (natAddLEFunctor h) R₂).obj' 1 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_4)
h₃ :
  IsIso (X := R₁.obj' (k₀ + 3) ⋯) (Y := R₂.obj' (k₀ + 3) ⋯)
    (app' (Functor.map (natAddLEFunctor h) φ) 3 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_6 :
      (Functor.obj (natAddLEFunctor h) R₁).obj' 3 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_6 ⟶
        (Functor.obj (natAddLEFunctor h) R₂).obj' 3 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_6)
h₄ :
  Mono (X := R₁.obj' (k₀ + 4) ⋯) (Y := R₂.obj' (k₀ + 4) ⋯)
    (app' (Functor.map (natAddLEFunctor h) φ) 4 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_7 :
      (Functor.obj (natAddLEFunctor h) R₁).obj' 4 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_7 ⟶
        (Functor.obj (natAddLEFunctor h) R₂).obj' 4 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_7)
h₀ :
  Epi (X := R₁.obj' (k₀ + 0) ⋯) (Y := R₂.obj' (k₀ + 0) ⋯)
    (app' (Functor.map (natAddLEFunctor h) φ) 0 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_2 :
      (Functor.obj (natAddLEFunctor h) R₁).obj' 0 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_2 ⟶
        (Functor.obj (natAddLEFunctor h) R₂).obj' 0 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_2)
⊢ IsIso (X := R₁.obj' (k₀ + 2) ⋯) (Y := R₂.obj' (k₀ + 2) ⋯)
    (app' (Functor.map (natAddLEFunctor h) φ) 2 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_9 :
      (Functor.obj (natAddLEFunctor h) R₁).obj' 2 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_9 ⟶
        (Functor.obj (natAddLEFunctor h) R₂).obj' 2 isIso_of_epi_of_isIso_of_isIso_of_mono._proof_9)

Tactic:
exact isIso_of_epi_of_isIso_of_isIso_of_mono (natAddLEFunctor_obj_exact h hR₁)
    (natAddLEFunctor_obj_exact h hR₂) _ h₀ h₁ h₃ h₄

Premises used: CategoryTheory.Abelian.isIso_of_epi_of_isIso_of_isIso_of_mono, CategoryTheory.ComposableArrows.natAddLEFunctor_obj_exact
-- Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
R₁ R₂ : ComposableArrows.{v_1, u_1} C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0
hR₁' : Epi (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_epi_mono'._proof_1)
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_2)
h₁ : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
⊢ (mk₃ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ := R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (X₃ := R₁.obj' 0 ⋯)
            (R₁.map' 0 1 ⋯ ⋯ :
              R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₁.map' 1 2 ⋯ ⋯ :
              R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            0).map'
        2 3 _proof_299✝ _proof_300✝ ≫
      𝟙
        ((mk₃ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
              R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := R₁.obj' 0 ⋯)
              (R₁.map' 0 1 ⋯ ⋯ :
                R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
              (R₁.map' 1 2 ⋯ ⋯ :
                R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
              0).obj'
          3 _proof_300✝) =
    app' φ 2 ⋯ ≫
      (mk₃ (X₀ := R₂.obj ⟨0, ⋯⟩) (X₁ := R₂.obj ⟨1, ⋯⟩) (X₃ := R₁.obj' 0 ⋯)
            (R₂.map' 0 1 ⋯ ⋯ : R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj ⟨1, ⋯⟩)
            (R₂.map' 1 2 ⋯ ⋯ : R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj ⟨2, ⋯⟩) 0).map'
        2 3 _proof_299✝ _proof_300✝

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
R₁ R₂ : ComposableArrows.{v_1, u_1} C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0
hR₁' : Epi (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_epi_mono'._proof_1)
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_2)
h₁ : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
ψ : mk₃ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ := R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    (X₃ := R₁.obj' 0 ⋯)
    (R₁.map' 0 1 ⋯ ⋯ : R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    (R₁.map' 1 2 ⋯ ⋯ : R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    0 ⟶
  mk₃ (X₀ := R₂.obj ⟨0, ⋯⟩) (X₁ := R₂.obj ⟨1, ⋯⟩) (X₃ := R₁.obj' 0 ⋯)
    (R₂.map' 0 1 ⋯ ⋯ : R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj ⟨1, ⋯⟩)
    (R₂.map' 1 2 ⋯ ⋯ : R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj ⟨2, ⋯⟩) 0 :=
  homMk₃ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯)
    (𝟙
      ((mk₃ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
            R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := R₁.obj' 0 ⋯)
            (R₁.map' 0 1 ⋯ ⋯ :
              R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₁.map' 1 2 ⋯ ⋯ :
              R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            0).obj'
        3 _proof_300✝))
    ⋯ ⋯ ⋯
dsimp :
  Mono.{v_1, u_1}
    (𝟙
      (Precomp.obj
        (mk₂ (X₀ := R₁.obj 1) (X₁ := R₁.obj 2) (X₂ := R₁.obj 0) (R₁.map (homOfLE ⋯) : R₁.obj 1 ⟶ R₁.obj 2)
          (0 : R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj' 0 ⋯))
        (R₁.obj 0) 3))
⊢ Mono (app' ψ 3 mono_of_epi_of_mono_of_mono'._proof_7)

Tactic:
dsimp [ψ]
-- Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
R₁ R₂ : ComposableArrows.{v_1, u_1} C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0
hR₁' : Epi (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_epi_mono'._proof_1)
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_2)
h₁ : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
ψ : mk₃ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ := R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    (X₃ := R₁.obj' 0 ⋯)
    (R₁.map' 0 1 ⋯ ⋯ : R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    (R₁.map' 1 2 ⋯ ⋯ : R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    0 ⟶
  mk₃ (X₀ := R₂.obj ⟨0, ⋯⟩) (X₁ := R₂.obj ⟨1, ⋯⟩) (X₃ := R₁.obj' 0 ⋯)
    (R₂.map' 0 1 ⋯ ⋯ : R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj ⟨1, ⋯⟩)
    (R₂.map' 1 2 ⋯ ⋯ : R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj ⟨2, ⋯⟩) 0 :=
  homMk₃ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯)
    (𝟙
      ((mk₃ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
            R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := R₁.obj' 0 ⋯)
            (R₁.map' 0 1 ⋯ ⋯ :
              R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₁.map' 1 2 ⋯ ⋯ :
              R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            0).obj'
        3 _proof_300✝))
    ⋯ ⋯ ⋯
⊢ Mono.{v_1, u_1}
    (𝟙
      (Precomp.obj
        (mk₂ (X₀ := R₁.obj 1) (X₁ := R₁.obj 2) (X₂ := R₁.obj 0) (R₁.map (homOfLE ⋯) : R₁.obj 1 ⟶ R₁.obj 2)
          (0 : R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj' 0 ⋯))
        (R₁.obj 0) 3))

Tactic:
infer_instance
-- Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
R₁ R₂ : ComposableArrows.{v_1, u_1} C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0
hR₁' : Epi (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_epi_mono'._proof_1)
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_2)
h₁ : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
ψ : mk₃ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ := R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    (X₃ := R₁.obj' 0 ⋯)
    (R₁.map' 0 1 ⋯ ⋯ : R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    (R₁.map' 1 2 ⋯ ⋯ : R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    0 ⟶
  mk₃ (X₀ := R₂.obj ⟨0, ⋯⟩) (X₁ := R₂.obj ⟨1, ⋯⟩) (X₃ := R₁.obj' 0 ⋯)
    (R₂.map' 0 1 ⋯ ⋯ : R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj ⟨1, ⋯⟩)
    (R₂.map' 1 2 ⋯ ⋯ : R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj ⟨2, ⋯⟩) 0 :=
  homMk₃ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯)
    (𝟙
      ((mk₃ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
            R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := R₁.obj' 0 ⋯)
            (R₁.map' 0 1 ⋯ ⋯ :
              R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₁.map' 1 2 ⋯ ⋯ :
              R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            0).obj'
        3 _proof_300✝))
    ⋯ ⋯ ⋯
⊢ Eq (α :=
    R₁.obj 0 ⟶
      Precomp.obj
        (mk₂ (X₀ := R₁.obj 1) (X₁ := R₁.obj 2) (X₂ := R₁.obj 0) (R₁.map (homOfLE ⋯) : R₁.obj 1 ⟶ R₁.obj 2)
          (0 : R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj' 0 ⋯))
        (R₁.obj 0) 2)
    (R₁.map
        (CategoryStruct.comp (Y := 1) (homOfLE ⋯ : Quiver.Hom (V := Fin 3) 0 1)
          (homOfLE ⋯ : Quiver.Hom (V := Fin 3) 1 2)) :
      R₁.obj 0 ⟶ R₁.obj 2)
    (0 :
      (mk₃ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ := R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
              (X₃ := R₁.obj' 0 ⋯)
              (R₁.map' 0 1 ⋯ ⋯ :
                R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
              (R₁.map' 1 2 ⋯ ⋯ :
                R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
              0).obj
          (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 3 1)) ⟶
        (mk₃ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
              R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := R₁.obj' 0 ⋯)
              (R₁.map' 0 1 ⋯ ⋯ :
                R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
              (R₁.map' 1 2 ⋯ ⋯ :
                R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
              0).obj
          ⟨2, ⋯⟩)

Tactic:
exact hR₁
-- Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
R₁ R₂ : ComposableArrows.{v_1, u_1} C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₁' : Epi (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_epi_mono'._proof_1)
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_2)
h₁ : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
⊢ R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0

Tactic:
simpa only [map'_comp R₁ 0 1 2] using hR₁.toIsComplex.zero 0

Premises used: CategoryTheory.ComposableArrows.Exact.toIsComplex, CategoryTheory.ComposableArrows.map'_comp, OfNat.ofNat, CategoryTheory.ComposableArrows.IsComplex.zero
-- Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
R₁ R₂ : ComposableArrows.{v_1, u_1} C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₀ : Epi (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
⊢ (mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
            R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (0 : R₁.obj' 0 ⋯ ⟶ R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₁.map' 0 1 ⋯ ⋯ :
              R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₁.map' 1 2 ⋯ ⋯ :
              R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))).map'
        0 1 homMk₁._proof_4 _proof_298✝ ≫
      app' φ 0 ⋯ =
    𝟙
        ((mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
              R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
              (0 : R₁.obj' 0 ⋯ ⟶ R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
              (R₁.map' 0 1 ⋯ ⋯ :
                R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
              (R₁.map' 1 2 ⋯ ⋯ :
                R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))).obj'
          0 _proof_297✝) ≫
      (mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
            R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (0 : R₁.obj' 0 ⋯ ⟶ R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₂.map' 0 1 ⋯ ⋯ :
              R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₂.map' 1 2 ⋯ ⋯ :
              R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))).map'
        0 1 homMk₁._proof_4 _proof_298✝

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
R₁ R₂ : ComposableArrows.{v_1, u_1} C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₀ : Epi (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
    R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (0 : R₁.obj' 0 ⋯ ⟶ R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    (R₁.map' 0 1 ⋯ ⋯ : R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    (R₁.map' 1 2 ⋯ ⋯ :
      R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) ⟶
  mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₂.obj ⟨0, ⋯⟩) (X₂ := R₂.obj ⟨1, ⋯⟩) (0 : R₁.obj' 0 ⋯ ⟶ R₂.obj ⟨0, ⋯⟩)
    (R₂.map' 0 1 ⋯ ⋯ : R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj ⟨1, ⋯⟩)
    (R₂.map' 1 2 ⋯ ⋯ : R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj ⟨2, ⋯⟩) :=
  homMk₃
    (𝟙
      ((mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
            R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (0 : R₁.obj' 0 ⋯ ⟶ R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₁.map' 0 1 ⋯ ⋯ :
              R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₁.map' 1 2 ⋯ ⋯ :
              R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))).obj'
        0 _proof_297✝))
    (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯ ⋯
dsimp : Epi.{v_1, u_1} (𝟙 (R₁.obj 0))
⊢ Epi (app' ψ 0 mono_of_epi_of_mono_of_mono'._proof_13)

Tactic:
dsimp [ψ]
-- Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
R₁ R₂ : ComposableArrows.{v_1, u_1} C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₀ : Epi (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
    R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (0 : R₁.obj' 0 ⋯ ⟶ R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    (R₁.map' 0 1 ⋯ ⋯ : R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    (R₁.map' 1 2 ⋯ ⋯ :
      R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) ⟶
  mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₂.obj ⟨0, ⋯⟩) (X₂ := R₂.obj ⟨1, ⋯⟩) (0 : R₁.obj' 0 ⋯ ⟶ R₂.obj ⟨0, ⋯⟩)
    (R₂.map' 0 1 ⋯ ⋯ : R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj ⟨1, ⋯⟩)
    (R₂.map' 1 2 ⋯ ⋯ : R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj ⟨2, ⋯⟩) :=
  homMk₃
    (𝟙
      ((mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
            R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (0 : R₁.obj' 0 ⋯ ⟶ R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₁.map' 0 1 ⋯ ⋯ :
              R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₁.map' 1 2 ⋯ ⋯ :
              R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))).obj'
        0 _proof_297✝))
    (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯ ⋯
⊢ Epi.{v_1, u_1} (𝟙 (R₁.obj 0))

Tactic:
infer_instance
-- Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
R₁ R₂ : ComposableArrows.{v_1, u_1} C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₀ : Epi (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
⊢ R₂.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0

Tactic:
simpa only [map'_comp R₂ 0 1 2] using hR₂.toIsComplex.zero 0

Premises used: CategoryTheory.ComposableArrows.Exact.toIsComplex, CategoryTheory.ComposableArrows.map'_comp, OfNat.ofNat, CategoryTheory.ComposableArrows.IsComplex.zero
-- Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
R₁ R₂ : ComposableArrows.{v_1, u_1} C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₀ : Mono (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_2)
h₁ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
⊢ (mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
            R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (0 : R₁.obj' 0 ⋯ ⟶ R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₁.map' 0 1 ⋯ ⋯ :
              R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₁.map' 1 2 ⋯ ⋯ :
              R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))).map'
        0 1 homMk₁._proof_4 _proof_298✝ ≫
      app' φ 0 ⋯ =
    𝟙
        ((mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
              R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
              (0 : R₁.obj' 0 ⋯ ⟶ R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
              (R₁.map' 0 1 ⋯ ⋯ :
                R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
              (R₁.map' 1 2 ⋯ ⋯ :
                R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))).obj'
          0 _proof_297✝) ≫
      (mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
            R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (0 : R₁.obj' 0 ⋯ ⟶ R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₂.map' 0 1 ⋯ ⋯ :
              R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₂.map' 1 2 ⋯ ⋯ :
              R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))).map'
        0 1 homMk₁._proof_4 _proof_298✝

Tactic:
simp