awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.CommSq (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X₁ X₂ X₃ X₄ : C
t : X₁ ⟶ X₂
l : X₁ ⟶ X₃
r : X₂ ⟶ X₄
b : X₃ ⟶ X₄
sq : IsPushout t l r b
A₀ : C
z : A₀ ⟶ kernel b
A₁ : C
π₁ : A₁ ⟶ A₀
w : Epi π₁
x₁ :
  A₁ ⟶
    ShortComplex.X₁
      (ShortComplex.mk (X₂ := biprod.{v, u} X₂ X₃) (X₃ :=
        Cocone.pt (F := parallelPair (Y := biprod.{v, u} X₂ X₃) (biprod.lift t (-l)) 0) ⋯.cokernelCofork)
        (biprod.lift t (-l)) (Cofork.π ⋯.cokernelCofork) ⋯)
hx₁ :
  π₁ ≫ z ≫ kernel.ι b ≫ biprod.inr =
    x₁ ≫
      ShortComplex.f
        (ShortComplex.mk (X₂ := biprod.{v, u} X₂ X₃) (X₃ :=
          Cocone.pt (F := parallelPair (Y := biprod.{v, u} X₂ X₃) (biprod.lift t (-l)) 0) ⋯.cokernelCofork)
          (biprod.lift t (-l)) (Cofork.π ⋯.cokernelCofork) ⋯)
⊢ ∃ (A' : C) (π : A' ⟶ A₀) (_ : Epi π) (x : A' ⟶ kernel t), π ≫ z = x ≫ kernel.map t b l r ⋯
have refine_1 : x₁ ≫ t = 0
Tactic: refine ⟨A₁, π₁, inferInstance, -kernel.lift _ x₁ ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.CommSq (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X₁ X₂ X₃ X₄ : C
t : X₁ ⟶ X₂
l : X₁ ⟶ X₃
r : X₂ ⟶ X₄
b : X₃ ⟶ X₄
sq : IsPushout t l r b
A₀ : C
z : A₀ ⟶ kernel b
A₁ : C
π₁ : A₁ ⟶ A₀
w : Epi π₁
x₁ :
  A₁ ⟶
    ShortComplex.X₁
      (ShortComplex.mk (X₂ := biprod.{v, u} X₂ X₃) (X₃ :=
        Cocone.pt (F := parallelPair (Y := biprod.{v, u} X₂ X₃) (biprod.lift t (-l)) 0) ⋯.cokernelCofork)
        (biprod.lift t (-l)) (Cofork.π ⋯.cokernelCofork) ⋯)
hx₁ :
  π₁ ≫ z ≫ kernel.ι b ≫ biprod.inr =
    x₁ ≫
      ShortComplex.f
        (ShortComplex.mk (X₂ := biprod.{v, u} X₂ X₃) (X₃ :=
          Cocone.pt (F := parallelPair (Y := biprod.{v, u} X₂ X₃) (biprod.lift t (-l)) 0) ⋯.cokernelCofork)
          (biprod.lift t (-l)) (Cofork.π ⋯.cokernelCofork) ⋯)
refine_1 : x₁ ≫ t = 0
⊢ ∃ (A' : C) (π : A' ⟶ A₀) (_ : Epi π) (x : A' ⟶ kernel t), π ≫ z = x ≫ kernel.map t b l r ⋯
have refine_2 : π₁ ≫ z = CategoryStruct.comp (Y := kernel t) (-kernel.lift t x₁ sorry) (kernel.map t b l r sorry)
Tactic: refine ⟨A₁, π₁, inferInstance, -kernel.lift _ x₁ ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.CommSq (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X₁ X₂ X₃ X₄ : C
t : X₁ ⟶ X₂
l : X₁ ⟶ X₃
r : X₂ ⟶ X₄
b : X₃ ⟶ X₄
sq : IsPushout t l r b
A₀ : C
z : A₀ ⟶ kernel b
A₁ : C
π₁ : A₁ ⟶ A₀
w : Epi π₁
x₁ :
  A₁ ⟶
    ShortComplex.X₁
      (ShortComplex.mk (X₂ := biprod.{v, u} X₂ X₃) (X₃ :=
        Cocone.pt (F := parallelPair (Y := biprod.{v, u} X₂ X₃) (biprod.lift t (-l)) 0) ⋯.cokernelCofork)
        (biprod.lift t (-l)) (Cofork.π ⋯.cokernelCofork) ⋯)
hx₁ :
  π₁ ≫ z ≫ kernel.ι b ≫ biprod.inr =
    x₁ ≫
      ShortComplex.f
        (ShortComplex.mk (X₂ := biprod.{v, u} X₂ X₃) (X₃ :=
          Cocone.pt (F := parallelPair (Y := biprod.{v, u} X₂ X₃) (biprod.lift t (-l)) 0) ⋯.cokernelCofork)
          (biprod.lift t (-l)) (Cofork.π ⋯.cokernelCofork) ⋯)
⊢ π₁ ≫ z = CategoryStruct.comp (Y := kernel t) (-kernel.lift t x₁ ⋯) (kernel.map t b l r ⋯)
have refine_2 : (π₁ ≫ z) ≫ equalizer.ι b 0 =
  CategoryStruct.comp (Y := equalizer b 0)
    (CategoryStruct.comp (Y := kernel t) (-kernel.lift t x₁ sorry) (kernel.map t b l r sorry)) (equalizer.ι b 0)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four (goal)
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 3
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
hR₁' :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₁ : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11)
h₃ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
⊢ Mono (app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4)
have h : ∀ {P : C} (g : P ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4),
  g ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4 = 0 → g = 0
Tactic: apply mono_of_cancel_zero
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four (goal)
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 3
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
hR₁' :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₁_1 : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11)
h₃ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
f₂ : A ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
⊢ f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
have rw : (f₂ ≫ R₁.map' 2 3 sorry sorry) ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 =
  0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7
Tactic: rw [← cancel_mono (app' φ 3 _), assoc, NatTrans.naturality, reassoc_of% h₁,
      zero_comp, zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four (goal)
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 3
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
hR₁' :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₁_1 : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11)
h₃ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
f₂ : A ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
rw :
  (f₂ ≫ R₁.map' 2 3 ⋯ ⋯) ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 =
    0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7
⊢ f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
have rw₁ : f₂ ≫ R₁.map' 2 3 sorry sorry ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 =
  0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7
Tactic: rw [← cancel_mono (app' φ 3 _), assoc, NatTrans.naturality, reassoc_of% h₁,
      zero_comp, zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four (goal)
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 3
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
hR₁' :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₁_1 : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11)
h₃ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
f₂ : A ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
rw :
  (f₂ ≫ R₁.map' 2 3 ⋯ ⋯) ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 =
    0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7
rw₁ :
  f₂ ≫ R₁.map' 2 3 ⋯ ⋯ ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 =
    0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7
⊢ f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
have rw₂ : Eq (α := A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
  (f₂ ≫ φ.app (⟨2, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 3 1)) ≫ R₂.map (homOfLE sorry) :
    A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
  (0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 : A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
Tactic: rw [← cancel_mono (app' φ 3 _), assoc, NatTrans.naturality, reassoc_of% h₁,
      zero_comp, zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four (goal)
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 3
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
hR₁' :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₁_1 : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11)
h₃ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
f₂ : A ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
rw :
  (f₂ ≫ R₁.map' 2 3 ⋯ ⋯) ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 =
    0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7
rw₁ :
  f₂ ≫ R₁.map' 2 3 ⋯ ⋯ ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 =
    0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7
rw₂ :
  Eq (α := A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
    (f₂ ≫ φ.app (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 3 1)) ≫ R₂.map (homOfLE ⋯) :
      A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
    (0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 : A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
⊢ f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
have rw₃ : Eq (α := A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
  (CategoryStruct.comp (Y := R₂.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4)
      (0 : A ⟶ R₂.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₂.map (homOfLE sorry) : R₂.obj (⟨2, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 3 1)) ⟶ R₂.obj ⟨3, sorry⟩) :
    A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
  (0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 : A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
Tactic: rw [← cancel_mono (app' φ 3 _), assoc, NatTrans.naturality, reassoc_of% h₁,
      zero_comp, zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four (goal)
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 3
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
hR₁' :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₁_1 : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11)
h₃ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
f₂ : A ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
rw :
  (f₂ ≫ R₁.map' 2 3 ⋯ ⋯) ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 =
    0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7
rw₁ :
  f₂ ≫ R₁.map' 2 3 ⋯ ⋯ ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 =
    0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7
rw₂ :
  Eq (α := A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
    (f₂ ≫ φ.app (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 3 1)) ≫ R₂.map (homOfLE ⋯) :
      A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
    (0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 : A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
rw₃ :
  Eq (α := A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
    (CategoryStruct.comp (Y := R₂.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4)
        (0 : A ⟶ R₂.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4)
        (R₂.map (homOfLE ⋯) : R₂.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 3 1)) ⟶ R₂.obj ⟨3, ⋯⟩) :
      A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
    (0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 : A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
⊢ f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
have rw₄ : Eq (α := A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7) 0 (0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
Tactic: rw [← cancel_mono (app' φ 3 _), assoc, NatTrans.naturality, reassoc_of% h₁,
      zero_comp, zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four (goal)
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 3
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
hR₁' :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₁_1 : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11)
h₃ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
f₂ : A ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
rw :
  (f₂ ≫ R₁.map' 2 3 ⋯ ⋯) ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 =
    0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7
rw₁ :
  f₂ ≫ R₁.map' 2 3 ⋯ ⋯ ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 =
    0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7
rw₂ :
  Eq (α := A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
    (f₂ ≫ φ.app (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 3 1)) ≫ R₂.map (homOfLE ⋯) :
      A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
    (0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 : A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
rw₃ :
  Eq (α := A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
    (CategoryStruct.comp (Y := R₂.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4)
        (0 : A ⟶ R₂.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4)
        (R₂.map (homOfLE ⋯) : R₂.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 3 1)) ⟶ R₂.obj ⟨3, ⋯⟩) :
      A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
    (0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7 : A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7)
rw₄ :
  Eq (α := A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7) 0 (0 ≫ app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
⊢ f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
have rw₅ : Eq.{v_1 + 1} (α := A ⟶ R₂.obj' 3 mono_of_epi_of_mono_of_mono'._proof_7) 0 0
Tactic: rw [← cancel_mono (app' φ 3 _), assoc, NatTrans.naturality, reassoc_of% h₁,
      zero_comp, zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four (hypothesis)
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 3
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
hR₁' :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₁_1 : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11)
h₃ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
f₂ : A ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
⊢ f₂ = 0
have h₂ : f₂ ≫ R₁.map' 2 3 sorry sorry = 0
Tactic: have h₂ : f₂ ≫ R₁.map' 2 3 = 0 := by
    rw [← cancel_mono (app' φ 3 _), assoc, NatTrans.naturality, reassoc_of% h₁,
      zero_comp, zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four (goal)
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 3
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
hR₁' :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₁_1 : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11)
h₃ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
f₂ : A ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
h₂ : f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
⊢ f₂ = 0
have h : ∀ (A₁ : C) (π₁ : A₁ ⟶ A),
  Epi π₁ →
    ∀
      (f₁ :
        A₁ ⟶
          ((mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
                  (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).sc
              sorry 0 sorry).X₁)
      (hf₁ :
        π₁ ≫ f₂ =
          f₁ ≫
            ((mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
                    (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).sc
                sorry 0 sorry).f),
      f₂ = 0
Tactic: obtain ⟨A₁, π₁, _, f₁, hf₁⟩ := (hR₁'.exact 0).exact_up_to_refinements f₂ h₂
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four (goal)
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 3
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
hR₁' :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₁_1 : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11)
h₃ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
f₂ : A ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
h₂ : f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
A₁ : C
π₁ : A₁ ⟶ A
w : Epi π₁
f₁ :
  A₁ ⟶
    ((mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
            (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).sc
        ⋯ 0 ⋯).X₁
hf₁ : π₁ ≫ f₂ = f₁ ≫ R₁.map (homOfLE ⋯)
⊢ Eq (α := A₁ ⟶ R₂.obj ⟨2, ⋯⟩)
    (CategoryStruct.comp (Y := R₂.obj' 1 ⋯) (f₁ ≫ app' φ 1 ⋯) (R₂.map' 1 2 ⋯ ⋯) : A₁ ⟶ R₂.obj ⟨2, ⋯⟩)
    (0 : A₁ ⟶ R₂.obj ⟨2, ⋯⟩)
have rw : Eq (α := A₁ ⟶ R₂.obj ⟨2, sorry⟩)
  (f₁ ≫ CategoryStruct.comp (Y := R₂.obj' 1 sorry) (app' φ 1 sorry) (R₂.map' 1 2 sorry sorry) : A₁ ⟶ R₂.obj ⟨2, sorry⟩)
  (0 : A₁ ⟶ R₂.obj ⟨2, sorry⟩)
Tactic: rw [assoc, ← NatTrans.naturality, ← reassoc_of% hf₁, h₁, comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four (goal)
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 3
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
hR₁' :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₁_1 : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11)
h₃ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
f₂ : A ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
h₂ : f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
A₁ : C
π₁ : A₁ ⟶ A
w : Epi π₁
f₁ :
  A₁ ⟶
    ((mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
            (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).sc
        ⋯ 0 ⋯).X₁
hf₁ : π₁ ≫ f₂ = f₁ ≫ R₁.map (homOfLE ⋯)
rw :
  Eq (α := A₁ ⟶ R₂.obj ⟨2, ⋯⟩)
    (f₁ ≫ CategoryStruct.comp (Y := R₂.obj' 1 ⋯) (app' φ 1 ⋯) (R₂.map' 1 2 ⋯ ⋯) : A₁ ⟶ R₂.obj ⟨2, ⋯⟩)
    (0 : A₁ ⟶ R₂.obj ⟨2, ⋯⟩)
⊢ Eq (α := A₁ ⟶ R₂.obj ⟨2, ⋯⟩)
    (CategoryStruct.comp (Y := R₂.obj' 1 ⋯) (f₁ ≫ app' φ 1 ⋯) (R₂.map' 1 2 ⋯ ⋯) : A₁ ⟶ R₂.obj ⟨2, ⋯⟩)
    (0 : A₁ ⟶ R₂.obj ⟨2, ⋯⟩)
have rw₁ : Eq (α := A₁ ⟶ R₂.obj ⟨2, sorry⟩)
  (f₁ ≫
      CategoryStruct.comp (Y := R₁.obj ⟨2, sorry⟩)
        (R₁.map (homOfLE sorry) : R₁.obj (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 3 1)) ⟶ R₁.obj ⟨2, sorry⟩)
        (φ.app ⟨2, sorry⟩ : R₁.obj (⟨2, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 3 1)) ⟶ R₂.obj ⟨2, sorry⟩) :
    A₁ ⟶ R₂.obj ⟨2, sorry⟩)
  (0 : A₁ ⟶ R₂.obj ⟨2, sorry⟩)
Tactic: rw [assoc, ← NatTrans.naturality, ← reassoc_of% hf₁, h₁, comp_zero]