awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
X : C
n : ℕ
X_1 Y Z : C
f : X_1 ⟶ Y
f' : Y ⟶ Z
⊢ Eq.{w + 1} (α := AddCommGrpCat.of (Ext.{w, v, u} X X_1 n) ⟶ AddCommGrpCat.of (Ext.{w, v, u} X Z n))
    (AddCommGrpCat.ofHom (Ext.postcomp (Ext.mk₀ (f ≫ f')) X ⋯))
    (AddCommGrpCat.ofHom (Ext.postcomp (Ext.mk₀ f) X ⋯) ≫ AddCommGrpCat.ofHom (Ext.postcomp (Ext.mk₀ f') X ⋯))
have hf : ∀ (α : (↑(AddCommGrpCat.of (Ext.{w, v, u} X X_1 n)) : Type w)),
  Eq (α := (↑(AddCommGrpCat.of (Ext.{w, v, u} X Z n)) : Type w))
    ((AddCommGrpCat.Hom.hom (AddCommGrpCat.ofHom (Ext.postcomp (Ext.mk₀ (f ≫ f')) X sorry)) :
          (a : (↑(AddCommGrpCat.of (Ext X X_1 n)) : Type w)) → (↑(AddCommGrpCat.of (Ext X Z n)) : Type w))
        α :
      (↑(AddCommGrpCat.of (Ext.{w, v, u} X Z n)) : Type w))
    ((AddCommGrpCat.Hom.hom
            (AddCommGrpCat.ofHom (Ext.postcomp (Ext.mk₀ f) X sorry) ≫ AddCommGrpCat.ofHom (Ext.postcomp (Ext.mk₀ f') X sorry)) :
          (a : (↑(AddCommGrpCat.of (Ext X X_1 n)) : Type w)) → (↑(AddCommGrpCat.of (Ext X Z n)) : Type w))
        α :
      (↑(AddCommGrpCat.of (Ext.{w, v, u} X Z n)) : Type w))
Tactic: ext α
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
X : C
n : ℕ
X_1 Y Z : C
f : X_1 ⟶ Y
f' : Y ⟶ Z
α : (↑(AddCommGrpCat.of (Ext.{w, v, u} X X_1 n)) : Type w)
⊢ Eq (α := Ext.{w, v, u} X Z n) (Ext.comp α (Ext.mk₀ (f ≫ f')) ⋯) (Ext.comp (Ext.comp α (Ext.mk₀ f) ⋯) (Ext.mk₀ f') ⋯)
have hf : Eq (α := Ext.{w, v, u} X Z n) (Ext.comp α (Ext.comp (Ext.mk₀ f) (Ext.mk₀ f') sorry) sorry)
  (Ext.comp (Ext.comp α (Ext.mk₀ f) sorry) (Ext.mk₀ f') sorry)
Tactic: rw [← Ext.mk₀_comp_mk₀]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
X : C
n : ℕ
X_1 Y Z : C
f : X_1 ⟶ Y
f' : Y ⟶ Z
α : (↑(AddCommGrpCat.of (Ext.{w, v, u} X X_1 n)) : Type w)
⊢ Eq (α := Ext.{w, v, u} X Z n) (Ext.comp α (Ext.comp (Ext.mk₀ f) (Ext.mk₀ f') ⋯) ⋯)
    (Ext.comp (Ext.comp α (Ext.mk₀ f) ⋯) (Ext.mk₀ f') ⋯)
have hf : Eq (α := Ext.{w, v, u} X Z n) (Ext.comp (Ext.comp α (Ext.mk₀ f) sorry) (Ext.mk₀ f') sorry)
  (Ext.comp α (Ext.comp (Ext.mk₀ f) (Ext.mk₀ f') sorry) sorry)
Tactic: symm
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
X : C
n : ℕ
X_1 Y Z : C
f : X_1 ⟶ Y
f' : Y ⟶ Z
α : (↑(AddCommGrpCat.of (Ext.{w, v, u} X X_1 n)) : Type w)
⊢ Eq (α := Ext.{w, v, u} X Z n) (Ext.comp (Ext.comp α (Ext.mk₀ f) ⋯) (Ext.mk₀ f') ⋯)
    (Ext.comp α (Ext.comp (Ext.mk₀ f) (Ext.mk₀ f') ⋯) ⋯)
have hf : n + 0 + 0 = n
Tactic: apply Ext.comp_assoc
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
n : ℕ
X₁ X₂ : Cᵒᵖ
f : X₁ ⟶ X₂
Y₁ Y₂ : C
g : Y₁ ⟶ Y₂
⊢ Eq.{w + 1} (α :=
    Functor.obj.{v, w, u, w + 1} (extFunctorObj (Opposite.unop X₁) n) Y₁ ⟶
      Functor.obj.{v, w, u, w + 1} (extFunctorObj (Opposite.unop X₂) n) Y₂)
    (Functor.map (extFunctorObj (Opposite.unop X₁) n) g ≫
        AddCommGrpCat.ofHom
          (AddMonoidHom.mk' (fun (α : Ext.{w, v, u} (Opposite.unop X₁) Y₂ n) ↦ Ext.comp (Ext.mk₀ f.unop) α ⋯) ⋯) :
      Functor.obj.{v, w, u, w + 1} (extFunctorObj (Opposite.unop X₁) n) Y₁ ⟶
        Functor.obj.{v, w, u, w + 1} (extFunctorObj (Opposite.unop X₂) n) Y₂)
    (AddCommGrpCat.ofHom
          (AddMonoidHom.mk' (fun (α : Ext.{w, v, u} (Opposite.unop X₁) Y₁ n) ↦ Ext.comp (Ext.mk₀ f.unop) α ⋯) ⋯) ≫
        Functor.map (extFunctorObj (Opposite.unop X₂) n) g :
      Functor.obj.{v, w, u, w + 1} (extFunctorObj (Opposite.unop X₁) n) Y₁ ⟶
        Functor.obj.{v, w, u, w + 1} (extFunctorObj (Opposite.unop X₂) n) Y₂)
have hf : ∀ (α : (↑(Functor.obj.{v, w, u, w + 1} (extFunctorObj (Opposite.unop X₁) n) Y₁) : Type w)),
  Eq (α := (↑(Functor.obj.{v, w, u, w + 1} (extFunctorObj (Opposite.unop X₂) n) Y₂) : Type w))
    ((AddCommGrpCat.Hom.hom
            (Functor.map (extFunctorObj (Opposite.unop X₁) n) g ≫
              AddCommGrpCat.ofHom
                (AddMonoidHom.mk' (fun (α : Ext.{w, v, u} (Opposite.unop X₁) Y₂ n) ↦ Ext.comp (Ext.mk₀ f.unop) α sorry)
                  sorry)) :
          (a : (↑((extFunctorObj (Opposite.unop X₁) n).obj Y₁) : Type w)) →
            (↑((extFunctorObj (Opposite.unop X₂) n).obj Y₂) : Type w))
        α :
      (↑(Functor.obj.{v, w, u, w + 1} (extFunctorObj (Opposite.unop X₂) n) Y₂) : Type w))
    ((AddCommGrpCat.Hom.hom
            (AddCommGrpCat.ofHom
                (AddMonoidHom.mk' (fun (α : Ext.{w, v, u} (Opposite.unop X₁) Y₁ n) ↦ Ext.comp (Ext.mk₀ f.unop) α sorry) sorry) ≫
              Functor.map (extFunctorObj (Opposite.unop X₂) n) g) :
          (a : (↑((extFunctorObj (Opposite.unop X₁) n).obj Y₁) : Type w)) →
            (↑((extFunctorObj (Opposite.unop X₂) n).obj Y₂) : Type w))
        α :
      (↑(Functor.obj.{v, w, u, w + 1} (extFunctorObj (Opposite.unop X₂) n) Y₂) : Type w))
Tactic: ext α
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
n : ℕ
X₁ X₂ : Cᵒᵖ
f : X₁ ⟶ X₂
Y₁ Y₂ : C
g : Y₁ ⟶ Y₂
α : (↑(Functor.obj.{v, w, u, w + 1} (extFunctorObj (Opposite.unop X₁) n) Y₁) : Type w)
⊢ Eq (α := Ext.{w, v, u} (Opposite.unop X₂) Y₂ n) (Ext.comp (Ext.mk₀ f.unop) (Ext.comp α (Ext.mk₀ g) ⋯) ⋯)
    (Ext.comp (Ext.comp (Ext.mk₀ f.unop) α ⋯) (Ext.mk₀ g) ⋯)
have hf : Eq (α := Ext.{w, v, u} (Opposite.unop X₂) Y₂ n) (Ext.comp (Ext.comp (Ext.mk₀ f.unop) α sorry) (Ext.mk₀ g) sorry)
  (Ext.comp (Ext.mk₀ f.unop) (Ext.comp α (Ext.mk₀ g) sorry) sorry)
Tactic: symm
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
n : ℕ
X₁ X₂ : Cᵒᵖ
f : X₁ ⟶ X₂
Y₁ Y₂ : C
g : Y₁ ⟶ Y₂
α : (↑(Functor.obj.{v, w, u, w + 1} (extFunctorObj (Opposite.unop X₁) n) Y₁) : Type w)
⊢ Eq (α := Ext.{w, v, u} (Opposite.unop X₂) Y₂ n) (Ext.comp (Ext.comp (Ext.mk₀ f.unop) α ⋯) (Ext.mk₀ g) ⋯)
    (Ext.comp (Ext.mk₀ f.unop) (Ext.comp α (Ext.mk₀ g) ⋯) ⋯)
have hf : 0 + n + 0 = n
Tactic: apply Ext.comp_assoc
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
n : ℕ
X₁ X₂ X₃ : Cᵒᵖ
f : X₁ ⟶ X₂
f' : X₂ ⟶ X₃
⊢ Eq (α :=
    Quiver.Hom (V := Functor.{v, w, u, w + 1} C AddCommGrpCat) (extFunctorObj (Opposite.unop X₁) n)
      (extFunctorObj (Opposite.unop X₃) n))
    {
      app := fun (Y : C) ↦
        AddCommGrpCat.ofHom
          (AddMonoidHom.mk' (fun (α : Ext.{w, v, u} (Opposite.unop X₁) Y n) ↦ Ext.comp (Ext.mk₀ (f ≫ f').unop) α ⋯) ⋯),
      naturality := ⋯ }
    (CategoryStruct.comp (Y := extFunctorObj (Opposite.unop X₂) n)
      {
        app := fun (Y : C) ↦
          AddCommGrpCat.ofHom
            (AddMonoidHom.mk' (fun (α : Ext.{w, v, u} (Opposite.unop X₁) Y n) ↦ Ext.comp (Ext.mk₀ f.unop) α ⋯) ⋯),
        naturality := ⋯ }
      {
        app := fun (Y : C) ↦
          AddCommGrpCat.ofHom
            (AddMonoidHom.mk' (fun (α : Ext.{w, v, u} (Opposite.unop X₂) Y n) ↦ Ext.comp (Ext.mk₀ f'.unop) α ⋯) ⋯),
        naturality := ⋯ })
have w : ∀ (Y : C) (α : (↑(Functor.obj.{v, w, u, w + 1} (extFunctorObj (Opposite.unop X₁) n) Y) : Type w)),
  Eq (α := (↑(Functor.obj.{v, w, u, w + 1} (extFunctorObj (Opposite.unop X₃) n) Y) : Type w))
    ((AddCommGrpCat.Hom.hom
            (NatTrans.app
              {
                app := fun (Y : C) ↦
                  AddCommGrpCat.ofHom
                    (AddMonoidHom.mk'
                      (fun (α : Ext.{w, v, u} (Opposite.unop X₁) Y n) ↦ Ext.comp (Ext.mk₀ (f ≫ f').unop) α sorry) sorry),
                naturality := sorry }
              Y) :
          (a : (↑((extFunctorObj (Opposite.unop X₁) n).obj Y) : Type w)) →
            (↑((extFunctorObj (Opposite.unop X₃) n).obj Y) : Type w))
        α :
      (↑(Functor.obj.{v, w, u, w + 1} (extFunctorObj (Opposite.unop X₃) n) Y) : Type w))
    ((AddCommGrpCat.Hom.hom
            (NatTrans.app
              (CategoryStruct.comp (Y := extFunctorObj (Opposite.unop X₂) n)
                {
                  app := fun (Y : C) ↦
                    AddCommGrpCat.ofHom
                      (AddMonoidHom.mk' (fun (α : Ext.{w, v, u} (Opposite.unop X₁) Y n) ↦ Ext.comp (Ext.mk₀ f.unop) α sorry)
                        sorry),
                  naturality := sorry }
                {
                  app := fun (Y : C) ↦
                    AddCommGrpCat.ofHom
                      (AddMonoidHom.mk'
                        (fun (α : Ext.{w, v, u} (Opposite.unop X₂) Y n) ↦ Ext.comp (Ext.mk₀ f'.unop) α sorry) sorry),
                  naturality := sorry })
              Y) :
          (a : (↑((extFunctorObj (Opposite.unop X₁) n).obj Y) : Type w)) →
            (↑((extFunctorObj (Opposite.unop X₃) n).obj Y) : Type w))
        α :
      (↑(Functor.obj.{v, w, u, w + 1} (extFunctorObj (Opposite.unop X₃) n) Y) : Type w))
Tactic: ext Y α
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
J : Type u_1
inst_3 : Fintype J
X : J → C
c : Bicone.{u_1, v, u} X
hc : c.IsBilimit
Y : C
n : ℕ
x : Ext.{w, v, u} c.pt Y n
⊢ (fun (e : (i : J) → Ext.{w, v, u} (X i) Y n) ↦ (∑ i : J, comp (mk₀ (c.π i)) (e i) ⋯ : Ext.{w, v, u} c.pt Y n))
      ((fun (e : Ext.{w, v, u} c.pt Y n) (i : J) ↦ comp (mk₀ (c.ι i)) e ⋯) x) =
    x
have simp : ∑ x_1 : J, comp (mk₀ (c.π x_1 ≫ c.ι x_1)) x sorry = x
Tactic: simp only [← comp_assoc_of_second_deg_zero, mk₀_comp_mk₀]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
J : Type u_1
inst_3 : Fintype J
X : J → C
c : Bicone.{u_1, v, u} X
hc : c.IsBilimit
Y : C
n : ℕ
x : Ext.{w, v, u} c.pt Y n
⊢ ∑ x_1 : J, comp (mk₀ (c.π x_1 ≫ c.ι x_1)) x ⋯ = x
have rw : comp (∑ i : J, mk₀ (c.π i ≫ c.ι i)) x sorry = x
Tactic: rw [← Ext.sum_comp, ← Ext.mk₀_sum, IsBilimit.total hc, mk₀_id_comp]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
J : Type u_1
inst_3 : Fintype J
X : J → C
c : Bicone.{u_1, v, u} X
hc : c.IsBilimit
Y : C
n : ℕ
x : Ext.{w, v, u} c.pt Y n
rw : comp (∑ i : J, mk₀ (c.π i ≫ c.ι i)) x ⋯ = x
⊢ ∑ x_1 : J, comp (mk₀ (c.π x_1 ≫ c.ι x_1)) x ⋯ = x
have rw₁ : comp (mk₀ (∑ i : J, c.π i ≫ c.ι i)) x sorry = x
Tactic: rw [← Ext.sum_comp, ← Ext.mk₀_sum, IsBilimit.total hc, mk₀_id_comp]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
J : Type u_1
inst_3 : Fintype J
X : J → C
c : Bicone.{u_1, v, u} X
hc : c.IsBilimit
Y : C
n : ℕ
x : Ext.{w, v, u} c.pt Y n
rw : comp (∑ i : J, mk₀ (c.π i ≫ c.ι i)) x ⋯ = x
rw₁ : comp (mk₀ (∑ i : J, c.π i ≫ c.ι i)) x ⋯ = x
⊢ ∑ x_1 : J, comp (mk₀ (c.π x_1 ≫ c.ι x_1)) x ⋯ = x
have rw₂ : comp (mk₀ (𝟙 c.pt)) x sorry = x
Tactic: rw [← Ext.sum_comp, ← Ext.mk₀_sum, IsBilimit.total hc, mk₀_id_comp]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
J : Type u_1
inst_3 : Fintype J
X : J → C
c : Bicone.{u_1, v, u} X
hc : c.IsBilimit
Y : C
n : ℕ
x : Ext.{w, v, u} c.pt Y n
rw : comp (∑ i : J, mk₀ (c.π i ≫ c.ι i)) x ⋯ = x
rw₁ : comp (mk₀ (∑ i : J, c.π i ≫ c.ι i)) x ⋯ = x
rw₂ : comp (mk₀ (𝟙 c.pt)) x ⋯ = x
⊢ ∑ x_1 : J, comp (mk₀ (c.π x_1 ≫ c.ι x_1)) x ⋯ = x
have rw₃ : x = x
Tactic: rw [← Ext.sum_comp, ← Ext.mk₀_sum, IsBilimit.total hc, mk₀_id_comp]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
J : Type u_1
inst_3 : Fintype J
X : J → C
c : Bicone.{u_1, v, u} X
hc : c.IsBilimit
Y : C
n : ℕ
x : (i : J) → Ext.{w, v, u} (X i) Y n
⊢ (fun (e : Ext.{w, v, u} c.pt Y n) (i : J) ↦ comp (mk₀ (c.ι i)) e ⋯)
      ((fun (e : (i : J) → Ext.{w, v, u} (X i) Y n) ↦ (∑ i : J, comp (mk₀ (c.π i)) (e i) ⋯ : Ext.{w, v, u} c.pt Y n))
        x) =
    x
have h : ∀ (i : J),
  (fun (e : Ext.{w, v, u} c.pt Y n) (i : J) ↦ comp (mk₀ (c.ι i)) e sorry)
      ((fun (e : (i : J) → Ext.{w, v, u} (X i) Y n) ↦ (∑ i : J, comp (mk₀ (c.π i)) (e i) sorry : Ext.{w, v, u} c.pt Y n)) x)
      i =
    x i
Tactic: ext i
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
J : Type u_1
inst_3 : Fintype J
X : J → C
c : Bicone.{u_1, v, u} X
hc : c.IsBilimit
Y : C
n : ℕ
x : (i : J) → Ext.{w, v, u} (X i) Y n
i : J
⊢ (fun (e : Ext.{w, v, u} c.pt Y n) (i : J) ↦ comp (mk₀ (c.ι i)) e ⋯)
      ((fun (e : (i : J) → Ext.{w, v, u} (X i) Y n) ↦ (∑ i : J, comp (mk₀ (c.π i)) (e i) ⋯ : Ext.{w, v, u} c.pt Y n)) x)
      i =
    x i
have h : ∑ x_1 : J, comp (mk₀ (c.ι i ≫ c.π x_1)) (x x_1) sorry = x i
Tactic: simp only [Ext.comp_sum, ← comp_assoc_of_second_deg_zero, mk₀_comp_mk₀]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
J : Type u_1
inst_3 : Fintype J
X : J → C
c : Bicone.{u_1, v, u} X
hc : c.IsBilimit
Y : C
n : ℕ
x : (i : J) → Ext.{w, v, u} (X i) Y n
i : J
⊢ ∑ x_1 : J, comp (mk₀ (c.ι i ≫ c.π x_1)) (x x_1) ⋯ = x i
have rw : ∀ (b : J), Membership.mem (γ := Finset J) Finset.univ b → b ≠ i → comp (mk₀ (c.ι i ≫ c.π b)) (x b) sorry = 0
Tactic: rw [Finset.sum_eq_single i _ (by simp), bicone_ι_π_self, mk₀_id_comp]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
J : Type u_1
inst_3 : Fintype J
X : J → C
c : Bicone.{u_1, v, u} X
hc : c.IsBilimit
Y : C
n : ℕ
x : (i : J) → Ext.{w, v, u} (X i) Y n
i b : J
h : Membership.mem (γ := Finset J) Finset.univ b
hij : b ≠ i
⊢ comp (mk₀ (c.ι i ≫ c.π b)) (x b) ⋯ = 0
have rw : Eq (α := Ext.{w, v, u} (X i) Y n) (comp (mk₀ (if h : i = b then eqToHom sorry else 0)) (x b) sorry) 0
Tactic: rw [c.ι_π, dif_neg hij.symm, mk₀_zero, zero_comp]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
J : Type u_1
inst_3 : Fintype J
X : J → C
c : Bicone.{u_1, v, u} X
hc : c.IsBilimit
Y : C
n : ℕ
x : (i : J) → Ext.{w, v, u} (X i) Y n
i b : J
h : Membership.mem (γ := Finset J) Finset.univ b
hij : b ≠ i
rw : Eq (α := Ext.{w, v, u} (X i) Y n) (comp (mk₀ (if h : i = b then eqToHom ⋯ else 0)) (x b) ⋯) 0
⊢ comp (mk₀ (c.ι i ≫ c.π b)) (x b) ⋯ = 0
have rw₁ : Eq (α := Ext.{w, v, u} (X i) Y n) (comp (mk₀ 0) (x b) sorry) 0
Tactic: rw [c.ι_π, dif_neg hij.symm, mk₀_zero, zero_comp]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
J : Type u_1
inst_3 : Fintype J
X : J → C
c : Bicone.{u_1, v, u} X
hc : c.IsBilimit
Y : C
n : ℕ
x : (i : J) → Ext.{w, v, u} (X i) Y n
i b : J
h : Membership.mem (γ := Finset J) Finset.univ b
hij : b ≠ i
rw : Eq (α := Ext.{w, v, u} (X i) Y n) (comp (mk₀ (if h : i = b then eqToHom ⋯ else 0)) (x b) ⋯) 0
rw₁ : Eq (α := Ext.{w, v, u} (X i) Y n) (comp (mk₀ 0) (x b) ⋯) 0
⊢ comp (mk₀ (c.ι i ≫ c.π b)) (x b) ⋯ = 0
have rw₂ : Eq (α := Ext.{w, v, u} (X i) Y n) (comp 0 (x b) sorry) 0
Tactic: rw [c.ι_π, dif_neg hij.symm, mk₀_zero, zero_comp]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
J : Type u_1
inst_3 : Fintype J
X : J → C
c : Bicone.{u_1, v, u} X
hc : c.IsBilimit
Y : C
n : ℕ
x : (i : J) → Ext.{w, v, u} (X i) Y n
i b : J
h : Membership.mem (γ := Finset J) Finset.univ b
hij : b ≠ i
rw : Eq (α := Ext.{w, v, u} (X i) Y n) (comp (mk₀ (if h : i = b then eqToHom ⋯ else 0)) (x b) ⋯) 0
rw₁ : Eq (α := Ext.{w, v, u} (X i) Y n) (comp (mk₀ 0) (x b) ⋯) 0
rw₂ : Eq (α := Ext.{w, v, u} (X i) Y n) (comp 0 (x b) ⋯) 0
⊢ comp (mk₀ (c.ι i ≫ c.π b)) (x b) ⋯ = 0
have rw₃ : Eq.{w + 1} (α := Ext.{w, v, u} (X i) Y n) 0 0
Tactic: rw [c.ι_π, dif_neg hij.symm, mk₀_zero, zero_comp]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
X : C
J : Type u_1
inst_3 : Fintype J
Y : J → C
c : Bicone.{u_1, v, u} Y
hc : c.IsBilimit
n : ℕ
x : (i : J) → Ext.{w, v, u} X (Y i) n
⊢ (fun (e : Ext.{w, v, u} X c.pt n) (i : J) ↦ e.comp (mk₀ (c.π i)) ⋯)
      ((fun (e : (i : J) → Ext.{w, v, u} X (Y i) n) ↦ (∑ i : J, (e i).comp (mk₀ (c.ι i)) ⋯ : Ext.{w, v, u} X c.pt n))
        x) =
    x
have h : ∀ (i : J),
  (fun (e : Ext.{w, v, u} X c.pt n) (i : J) ↦ e.comp (mk₀ (c.π i)) sorry)
      ((fun (e : (i : J) → Ext.{w, v, u} X (Y i) n) ↦ (∑ i : J, (e i).comp (mk₀ (c.ι i)) sorry : Ext.{w, v, u} X c.pt n)) x)
      i =
    x i
Tactic: ext i
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
X : C
J : Type u_1
inst_3 : Fintype J
Y : J → C
c : Bicone.{u_1, v, u} Y
hc : c.IsBilimit
n : ℕ
x : (i : J) → Ext.{w, v, u} X (Y i) n
i : J
⊢ (fun (e : Ext.{w, v, u} X c.pt n) (i : J) ↦ e.comp (mk₀ (c.π i)) ⋯)
      ((fun (e : (i : J) → Ext.{w, v, u} X (Y i) n) ↦ (∑ i : J, (e i).comp (mk₀ (c.ι i)) ⋯ : Ext.{w, v, u} X c.pt n)) x)
      i =
    x i
have h : ∑ x_1 : J, (x x_1).comp (mk₀ (c.ι x_1 ≫ c.π i)) sorry = x i
Tactic: simp only [Ext.sum_comp, comp_assoc_of_second_deg_zero, mk₀_comp_mk₀]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
X : C
J : Type u_1
inst_3 : Fintype J
Y : J → C
c : Bicone.{u_1, v, u} Y
hc : c.IsBilimit
n : ℕ
x : (i : J) → Ext.{w, v, u} X (Y i) n
i : J
⊢ ∑ x_1 : J, (x x_1).comp (mk₀ (c.ι x_1 ≫ c.π i)) ⋯ = x i
have rw : ∀ (b : J), Membership.mem (γ := Finset J) Finset.univ b → b ≠ i → (x b).comp (mk₀ (c.ι b ≫ c.π i)) sorry = 0
Tactic: rw [Finset.sum_eq_single i _ (by simp), bicone_ι_π_self, comp_mk₀_id]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
X : C
J : Type u_1
inst_3 : Fintype J
Y : J → C
c : Bicone.{u_1, v, u} Y
hc : c.IsBilimit
n : ℕ
x : (i : J) → Ext.{w, v, u} X (Y i) n
i b : J
h : Membership.mem (γ := Finset J) Finset.univ b
hij : b ≠ i
⊢ (x b).comp (mk₀ (c.ι b ≫ c.π i)) ⋯ = 0
have rw : Eq (α := Ext.{w, v, u} X (Y i) n) ((x b).comp (mk₀ (if h : b = i then eqToHom sorry else 0)) sorry) 0
Tactic: rw [c.ι_π, dif_neg hij, mk₀_zero, comp_zero]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
X : C
J : Type u_1
inst_3 : Fintype J
Y : J → C
c : Bicone.{u_1, v, u} Y
hc : c.IsBilimit
n : ℕ
x : (i : J) → Ext.{w, v, u} X (Y i) n
i b : J
h : Membership.mem (γ := Finset J) Finset.univ b
hij : b ≠ i
rw : Eq (α := Ext.{w, v, u} X (Y i) n) ((x b).comp (mk₀ (if h : b = i then eqToHom ⋯ else 0)) ⋯) 0
⊢ (x b).comp (mk₀ (c.ι b ≫ c.π i)) ⋯ = 0
have rw₁ : Eq (α := Ext.{w, v, u} X (Y i) n) ((x b).comp (mk₀ 0) sorry) 0
Tactic: rw [c.ι_π, dif_neg hij, mk₀_zero, comp_zero]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
X : C
J : Type u_1
inst_3 : Fintype J
Y : J → C
c : Bicone.{u_1, v, u} Y
hc : c.IsBilimit
n : ℕ
x : (i : J) → Ext.{w, v, u} X (Y i) n
i b : J
h : Membership.mem (γ := Finset J) Finset.univ b
hij : b ≠ i
rw : Eq (α := Ext.{w, v, u} X (Y i) n) ((x b).comp (mk₀ (if h : b = i then eqToHom ⋯ else 0)) ⋯) 0
rw₁ : Eq (α := Ext.{w, v, u} X (Y i) n) ((x b).comp (mk₀ 0) ⋯) 0
⊢ (x b).comp (mk₀ (c.ι b ≫ c.π i)) ⋯ = 0
have rw₂ : Eq (α := Ext.{w, v, u} X (Y i) n) ((x b).comp 0 sorry) 0
Tactic: rw [c.ι_π, dif_neg hij, mk₀_zero, comp_zero]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
X : C
J : Type u_1
inst_3 : Fintype J
Y : J → C
c : Bicone.{u_1, v, u} Y
hc : c.IsBilimit
n : ℕ
x : (i : J) → Ext.{w, v, u} X (Y i) n
i b : J
h : Membership.mem (γ := Finset J) Finset.univ b
hij : b ≠ i
rw : Eq (α := Ext.{w, v, u} X (Y i) n) ((x b).comp (mk₀ (if h : b = i then eqToHom ⋯ else 0)) ⋯) 0
rw₁ : Eq (α := Ext.{w, v, u} X (Y i) n) ((x b).comp (mk₀ 0) ⋯) 0
rw₂ : Eq (α := Ext.{w, v, u} X (Y i) n) ((x b).comp 0 ⋯) 0
⊢ (x b).comp (mk₀ (c.ι b ≫ c.π i)) ⋯ = 0
have rw₃ : Eq.{w + 1} (α := Ext.{w, v, u} X (Y i) n) 0 0
Tactic: rw [c.ι_π, dif_neg hij, mk₀_zero, comp_zero]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
this : HasDerivedCategory C := HasDerivedCategory.standard C
⊢ HasExt.{w', v, u} C ↔ ∀ (X Y : C) (n : ℕ), Small.{w', w} (Ext.{w, v, u} X Y n)
have simp : (∀ (X Y : C) (n : ℤ),
    0 ≤ n →
      Small.{w', max u v}
        (Quiver.Hom (V := DerivedCategory C) (Functor.obj (singleFunctor C 0) X)
          (Functor.obj (shiftFunctor (DerivedCategory C) n) (Functor.obj (singleFunctor C 0) Y)))) ↔
  ∀ (X Y : C) (n : ℕ),
    Small.{w', max u v}
      (ShiftedHom (C := DerivedCategory C) (M := ℤ) (Functor.obj (singleFunctor C 0) X)
        (Functor.obj (singleFunctor C 0) Y) (↑n : ℤ))
Tactic: simp only [hasExt_iff, small_congr Ext.homEquiv]
-- Lean 4 proof step from Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
this : HasDerivedCategory C := HasDerivedCategory.standard C
⊢ (∀ (X Y : C) (n : ℤ),
      0 ≤ n →
        Small.{w', max u v}
          (Quiver.Hom (V := DerivedCategory C) (Functor.obj (singleFunctor C 0) X)
            (Functor.obj (shiftFunctor (DerivedCategory C) n) (Functor.obj (singleFunctor C 0) Y)))) ↔
    ∀ (X Y : C) (n : ℕ),
      Small.{w', max u v}
        (ShiftedHom (C := DerivedCategory C) (M := ℤ) (Functor.obj (singleFunctor C 0) X)
          (Functor.obj (singleFunctor C 0) Y) (↑n : ℤ))
have mp : (∀ (X Y : C) (n : ℤ),
    0 ≤ n →
      Small.{w', max u v}
        (Quiver.Hom (V := DerivedCategory C) (Functor.obj (singleFunctor C 0) X)
          (Functor.obj (shiftFunctor (DerivedCategory C) n) (Functor.obj (singleFunctor C 0) Y)))) →
  ∀ (X Y : C) (n : ℕ),
    Small.{w', max u v}
      (ShiftedHom (C := DerivedCategory C) (M := ℤ) (Functor.obj (singleFunctor C 0) X)
        (Functor.obj (singleFunctor C 0) Y) (↑n : ℤ))
Tactic: constructor