awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Indization (hypothesis)
Goal:
C : Type v
inst : SmallCategory C
inst_1 : Abelian C
X Y : Ind C
f : X ⟶ Y
I : Type v
w : SmallCategory I
w_1 : IsFiltered I
F G : I ⥤ C
ϕ : F ⟶ G
i : Arrow.mk f ≅ Arrow.mk (X := Functor.obj (Ind.lim I) F) (Functor.map (Ind.lim I) ϕ)
i' : Arrow.mk (X := Abelian.coimage f) (Y := Abelian.image f) (coimageImageComparison f) ≅
  Arrow.mk (X := Abelian.coimage (P := Functor.obj (Ind.lim I) F) (Functor.map (Ind.lim I) ϕ)) (Y :=
    Abelian.image (P := Functor.obj (Ind.lim I) F) (Functor.map (Ind.lim I) ϕ))
    (coimageImageComparison (Functor.map (Ind.lim I) ϕ)) :=
  Functor.mapIso coimageImageComparisonFunctor i
⊢ IsIso (X := Abelian.coimage f) (Y := Abelian.image f) (coimageImageComparison f)
have this : IsIso i'.hom
Tactic: have := Iso.isIso_hom i'
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Indization (goal)
Goal:
C : Type v
inst : SmallCategory C
inst_1 : Abelian C
X Y : Ind C
f : X ⟶ Y
I : Type v
w : SmallCategory I
w_1 : IsFiltered I
F G : I ⥤ C
ϕ : F ⟶ G
i : Arrow.mk f ≅ Arrow.mk (X := Functor.obj (Ind.lim I) F) (Functor.map (Ind.lim I) ϕ)
i' : Arrow.mk (X := Abelian.coimage f) (Y := Abelian.image f) (coimageImageComparison f) ≅
  Arrow.mk (X := Abelian.coimage (P := Functor.obj (Ind.lim I) F) (Functor.map (Ind.lim I) ϕ)) (Y :=
    Abelian.image (P := Functor.obj (Ind.lim I) F) (Functor.map (Ind.lim I) ϕ))
    (coimageImageComparison (Functor.map (Ind.lim I) ϕ)) :=
  Functor.mapIso coimageImageComparisonFunctor i
this : IsIso i'.hom
⊢ IsIso (X := Abelian.coimage f) (Y := Abelian.image f) (coimageImageComparison f)
have rw : IsIso (X := Functor.obj (Ind.lim I) (Abelian.coimage ϕ)) (Y := Functor.obj (Ind.lim I) (Abelian.image ϕ))
  (Functor.map (Ind.lim I) (coimageImageComparison ϕ))
Tactic: rw [Arrow.isIso_iff_isIso_of_isIso i'.hom,
    Arrow.isIso_iff_isIso_of_isIso (PreservesCoimageImageComparison.iso (Ind.lim I) ϕ).inv]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
J : C
hP : PreservesFiniteColimits.{v, v, u, v + 1} (D := ModuleCat (End J)) (preadditiveYonedaObj J)
⊢ Injective.{v, u} J
have rw : Functor.PreservesEpimorphisms.{v, v, u, v + 1} (D := ModuleCat (End J)) (preadditiveYonedaObj J)
Tactic: rw [injective_iff_preservesEpimorphisms_preadditive_yoneda_obj']
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Basic (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
J : C
hP : PreservesFiniteColimits.{v, v, u, v + 1} (D := ModuleCat (End J)) (preadditiveYonedaObj J)
⊢ Functor.PreservesEpimorphisms.{v, v, u, v + 1} (D := ModuleCat (End J)) (preadditiveYonedaObj J)
have this : Functor.PreservesHomology.{v, u, v, v + 1} (D := ModuleCat (End J)) (preadditiveYonedaObj J)
Tactic: have := Functor.preservesHomologyOfExact (preadditiveYonedaObj J)
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (hypothesis)
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
hX : HasInjectiveDimensionLT.{v, u} X n
i : ℕ
hi : n ≤ i
Y : C
this : HasExt C := ⋯
⊢ Subsingleton.{w + 1} (Ext.{w, v, u} Y X i)
have this₁ : ∀ ⦃Y : C⦄, Subsingleton (Ext Y X i)
Tactic: have := hX.subsingleton' i hi
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (hypothesis)
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
hX : ∀ (i : ℕ), n ≤ i → ∀ ⦃Y : C⦄ (e : Ext.{w, v, u} Y X i), e = 0
i : ℕ
hi : n ≤ i
Y : C
⊢ Subsingleton (Ext Y X i)
have this : Subsingleton.{w + 1} (Ext.{w, v, u} Y X i)
Tactic: have : Subsingleton (Ext Y X i) := ⟨fun e₁ e₂ ↦ by simp only [hX i hi]⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X : C
hX : IsZero.{v, u} X
this : HasExt C := ⋯
⊢ HasInjectiveDimensionLT.{v, u} X 0
have rw : ∀ (i : ℕ), 0 ≤ i → ∀ ⦃Y : C⦄ (e : Ext Y X i), e = 0
Tactic: rw [hasInjectiveDimensionLT_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X : C
hX : IsZero.{v, u} X
this : HasExt C := ⋯
i : ℕ
hi : 0 ≤ i
Y : C
e : Ext Y X i
⊢ e = 0
have rw : e.comp (Ext.mk₀ (𝟙 X)) sorry = 0
Tactic: rw [← e.comp_mk₀_id, hX.eq_zero_of_tgt (𝟙 X), Ext.mk₀_zero, Ext.comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X : C
hX : IsZero.{v, u} X
this : HasExt C := ⋯
i : ℕ
hi : 0 ≤ i
Y : C
e : Ext Y X i
rw : e.comp (Ext.mk₀ (𝟙 X)) ⋯ = 0
⊢ e = 0
have rw₁ : Eq (α := Ext Y X i) (e.comp (Ext.mk₀ 0) sorry) 0
Tactic: rw [← e.comp_mk₀_id, hX.eq_zero_of_tgt (𝟙 X), Ext.mk₀_zero, Ext.comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X : C
hX : IsZero.{v, u} X
this : HasExt C := ⋯
i : ℕ
hi : 0 ≤ i
Y : C
e : Ext Y X i
rw : e.comp (Ext.mk₀ (𝟙 X)) ⋯ = 0
rw₁ : Eq (α := Ext Y X i) (e.comp (Ext.mk₀ 0) ⋯) 0
⊢ e = 0
have rw₂ : Eq (α := Ext Y X i) (e.comp 0 sorry) 0
Tactic: rw [← e.comp_mk₀_id, hX.eq_zero_of_tgt (𝟙 X), Ext.mk₀_zero, Ext.comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X : C
hX : IsZero.{v, u} X
this : HasExt C := ⋯
i : ℕ
hi : 0 ≤ i
Y : C
e : Ext Y X i
rw : e.comp (Ext.mk₀ (𝟙 X)) ⋯ = 0
rw₁ : Eq (α := Ext Y X i) (e.comp (Ext.mk₀ 0) ⋯) 0
rw₂ : Eq (α := Ext Y X i) (e.comp 0 ⋯) 0
⊢ e = 0
have rw₃ : Eq (α := Ext Y X i) 0 0
Tactic: rw [← e.comp_mk₀_id, hX.eq_zero_of_tgt (𝟙 X), Ext.mk₀_zero, Ext.comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X : C
inst_2 : HasInjectiveDimensionLT.{v, u} X 0
this : HasExt C := ⋯
⊢ IsZero.{v, u} X
have rw : Eq.{v + 1} (α := X ⟶ X) (𝟙 X) 0
Tactic: rw [IsZero.iff_id_eq_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X : C
inst_2 : HasInjectiveDimensionLT.{v, u} X 0
this : HasExt C := ⋯
⊢ Eq.{v + 1} (α := X ⟶ X) (𝟙 X) 0
have a : Eq (α := Ext X X 0) ((Equiv.symm Ext.homEquiv₀ : (a : X ⟶ X) → Ext X X 0) (𝟙 X) : Ext X X 0)
  ((Equiv.symm Ext.homEquiv₀ : (a : X ⟶ X) → Ext X X 0) 0 : Ext X X 0)
Tactic: apply Ext.homEquiv₀.symm.injective
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X : C
n m : ℕ
h : n ≤ m
inst_2 : HasInjectiveDimensionLT.{v, u} X n
this : HasExt C := ⋯
⊢ HasInjectiveDimensionLT.{v, u} X m
have rw : ∀ (i : ℕ), m ≤ i → ∀ ⦃Y : C⦄ (e : Ext Y X i), e = 0
Tactic: rw [hasInjectiveDimensionLT_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X : C
n : ℕ
inst_2 : Injective.{v, u} X
this : HasExt C := ⋯
⊢ HasInjectiveDimensionLT.{v, u} X 1
have rw : ∀ (i : ℕ), 1 ≤ i → ∀ ⦃Y : C⦄ (e : Ext Y X i), e = 0
Tactic: rw [hasInjectiveDimensionLT_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X : C
n : ℕ
inst_2 : Injective.{v, u} X
this : HasExt C := ⋯
i : ℕ
hi : 1 ≤ i
Y : C
e : Ext Y X i
⊢ e = 0
have zero : ∀ (hi : LE.le (α := ℕ) 1 0) (e : Ext Y X 0), e = 0
Tactic: obtain _ | i := i
Cleared hypotheses: i, hi, e
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X : C
n : ℕ
inst_2 : Injective.{v, u} X
this : HasExt C := ⋯
i : ℕ
hi : 1 ≤ i
Y : C
e : Ext Y X i
zero : ∀ (hi : LE.le (α := ℕ) 1 0) (e : Ext Y X 0), e = 0
⊢ e = 0
have succ : ∀ (i : ℕ) (hi : 1 ≤ i + 1) (e : Ext Y X (i + 1)), e = 0
Tactic: obtain _ | i := i
Cleared hypotheses: i, hi, e
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X : C
inst_2 : HasExt.{w, v, u} C
⊢ Injective.{v, u} X ↔ ∀ ⦃Y : C⦄, Subsingleton.{w + 1} (Ext.{w, v, u} Y X 1)
have refine : ∀ (h : ∀ ⦃Y : C⦄, Subsingleton.{w + 1} (Ext.{w, v, u} Y X 1)) {X_1 Y : C} (f : X_1 ⟶ X) (g : X_1 ⟶ Y),
  Mono g → ∃ (h : Y ⟶ X), g ≫ h = f
Tactic: refine ⟨fun h ↦ HasInjectiveDimensionLT.subsingleton X 1 1 (by rfl),
    fun h ↦ ⟨fun f g _ ↦ ?_⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X : C
inst_2 : HasExt.{w, v, u} C
h : ∀ ⦃Y : C⦄, Subsingleton.{w + 1} (Ext.{w, v, u} Y X 1)
X_1 Y : C
f : X_1 ⟶ X
g : X_1 ⟶ Y
h_1 : Mono g
⊢ ∃ (h : Y ⟶ X), g ≫ h = f
have obtain : ∀
  (φ :
    Ext.{w, v, u}
      (ShortComplex.X₂
        { X₁ := X_1, X₂ := Y, X₃ := cokernel g, f := g, g := cokernel.π g, zero := sorry : ShortComplex.{v, u} C })
      X 0)
  (hφ :
    Ext.comp
        (Ext.mk₀
          (ShortComplex.f
            { X₁ := X_1, X₂ := Y, X₃ := cokernel g, f := g, g := cokernel.π g, zero := sorry : ShortComplex.{v, u} C }))
        φ sorry =
      Ext.mk₀ f),
  ∃ (h : Y ⟶ X), g ≫ h = f
Tactic: obtain ⟨φ, hφ⟩ := Ext.contravariant_sequence_exact₁ { exact := ShortComplex.exact_cokernel g } _
    (Ext.mk₀ f) (zero_add 1) (by subsingleton)
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X : C
inst_2 : HasExt.{w, v, u} C
h : ∀ ⦃Y : C⦄, Subsingleton.{w + 1} (Ext.{w, v, u} Y X 1)
X_1 Y : C
f : X_1 ⟶ X
g : X_1 ⟶ Y
h_1 : Mono g
φ :
  Ext.{w, v, u}
    (ShortComplex.X₂
      { X₁ := X_1, X₂ := Y, X₃ := cokernel g, f := g, g := cokernel.π g, zero := ⋯ : ShortComplex.{v, u} C })
    X 0
hφ :
  Ext.comp
      (Ext.mk₀
        (ShortComplex.f
          { X₁ := X_1, X₂ := Y, X₃ := cokernel g, f := g, g := cokernel.π g, zero := ⋯ : ShortComplex.{v, u} C }))
      φ ⋯ =
    Ext.mk₀ f
⊢ ∃ (h : Y ⟶ X), g ≫ h = f
have obtain : ∀
  (φ :
    ShortComplex.X₂
        { X₁ := X_1, X₂ := Y, X₃ := cokernel g, f := g, g := cokernel.π g, zero := sorry : ShortComplex.{v, u} C } ⟶
      X)
  (hφ :
    Eq.{w + 1} (α :=
      Ext.{w, v, u}
        (ShortComplex.X₁
          { X₁ := X_1, X₂ := Y, X₃ := cokernel g, f := g, g := cokernel.π g, zero := sorry : ShortComplex.{v, u} C })
        X 0)
      (Ext.comp
        (Ext.mk₀
          (ShortComplex.f
            { X₁ := X_1, X₂ := Y, X₃ := cokernel g, f := g, g := cokernel.π g, zero := sorry : ShortComplex.{v, u} C }))
        ((Equiv.symm Ext.homEquiv₀ :
            (a : { X₁ := X_1, X₂ := Y, X₃ := cokernel g, f := g, g := cokernel.π g, zero := sorry }.X₂ ⟶ X) →
              Ext { X₁ := X_1, X₂ := Y, X₃ := cokernel g, f := g, g := cokernel.π g, zero := sorry }.X₂ X 0)
          φ)
        sorry)
      (Ext.mk₀ f)),
  ∃ (h : Y ⟶ X), g ≫ h = f
Tactic: obtain ⟨φ, rfl⟩ := Ext.homEquiv₀.symm.surjective φ
Cleared hypotheses: φ, hφ
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y : C
h : Retract X Y
n : ℕ
inst_2 : HasInjectiveDimensionLT.{v, u} Y n
this : HasExt C := ⋯
⊢ HasInjectiveDimensionLT.{v, u} X n
have rw : ∀ (i : ℕ), n ≤ i → ∀ ⦃Y : C⦄ (e : Ext Y X i), e = 0
Tactic: rw [hasInjectiveDimensionLT_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y : C
h : Retract X Y
n : ℕ
inst_2 : HasInjectiveDimensionLT.{v, u} Y n
this : HasExt C := ⋯
i : ℕ
hi : n ≤ i
T : C
x : Ext T X i
⊢ x = 0
have rw : x.comp (Ext.mk₀ (𝟙 X)) sorry = 0
Tactic: rw [← x.comp_mk₀_id, ← h.retract, ← Ext.mk₀_comp_mk₀, ← Ext.comp_assoc_of_second_deg_zero,
    (x.comp (Ext.mk₀ h.i) (add_zero i)).eq_zero_of_hasInjectiveDimensionLT n hi, Ext.zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y : C
h : Retract X Y
n : ℕ
inst_2 : HasInjectiveDimensionLT.{v, u} Y n
this : HasExt C := ⋯
i : ℕ
hi : n ≤ i
T : C
x : Ext T X i
rw : x.comp (Ext.mk₀ (𝟙 X)) ⋯ = 0
⊢ x = 0
have rw₁ : x.comp (Ext.mk₀ (h.i ≫ h.r)) sorry = 0
Tactic: rw [← x.comp_mk₀_id, ← h.retract, ← Ext.mk₀_comp_mk₀, ← Ext.comp_assoc_of_second_deg_zero,
    (x.comp (Ext.mk₀ h.i) (add_zero i)).eq_zero_of_hasInjectiveDimensionLT n hi, Ext.zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y : C
h : Retract X Y
n : ℕ
inst_2 : HasInjectiveDimensionLT.{v, u} Y n
this : HasExt C := ⋯
i : ℕ
hi : n ≤ i
T : C
x : Ext T X i
rw : x.comp (Ext.mk₀ (𝟙 X)) ⋯ = 0
rw₁ : x.comp (Ext.mk₀ (h.i ≫ h.r)) ⋯ = 0
⊢ x = 0
have rw₂ : x.comp (Ext.comp (Ext.mk₀ h.i) (Ext.mk₀ h.r) sorry) sorry = 0
Tactic: rw [← x.comp_mk₀_id, ← h.retract, ← Ext.mk₀_comp_mk₀, ← Ext.comp_assoc_of_second_deg_zero,
    (x.comp (Ext.mk₀ h.i) (add_zero i)).eq_zero_of_hasInjectiveDimensionLT n hi, Ext.zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y : C
h : Retract X Y
n : ℕ
inst_2 : HasInjectiveDimensionLT.{v, u} Y n
this : HasExt C := ⋯
i : ℕ
hi : n ≤ i
T : C
x : Ext T X i
rw : x.comp (Ext.mk₀ (𝟙 X)) ⋯ = 0
rw₁ : x.comp (Ext.mk₀ (h.i ≫ h.r)) ⋯ = 0
rw₂ : x.comp (Ext.comp (Ext.mk₀ h.i) (Ext.mk₀ h.r) ⋯) ⋯ = 0
⊢ x = 0
have rw₃ : (x.comp (Ext.mk₀ h.i) sorry).comp (Ext.mk₀ h.r) sorry = 0
Tactic: rw [← x.comp_mk₀_id, ← h.retract, ← Ext.mk₀_comp_mk₀, ← Ext.comp_assoc_of_second_deg_zero,
    (x.comp (Ext.mk₀ h.i) (add_zero i)).eq_zero_of_hasInjectiveDimensionLT n hi, Ext.zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y : C
h : Retract X Y
n : ℕ
inst_2 : HasInjectiveDimensionLT.{v, u} Y n
this : HasExt C := ⋯
i : ℕ
hi : n ≤ i
T : C
x : Ext T X i
rw : x.comp (Ext.mk₀ (𝟙 X)) ⋯ = 0
rw₁ : x.comp (Ext.mk₀ (h.i ≫ h.r)) ⋯ = 0
rw₂ : x.comp (Ext.comp (Ext.mk₀ h.i) (Ext.mk₀ h.r) ⋯) ⋯ = 0
rw₃ : (x.comp (Ext.mk₀ h.i) ⋯).comp (Ext.mk₀ h.r) ⋯ = 0
⊢ x = 0
have rw₄ : Eq (α := Ext T X i) (Ext.comp 0 (Ext.mk₀ h.r) sorry) 0
Tactic: rw [← x.comp_mk₀_id, ← h.retract, ← Ext.mk₀_comp_mk₀, ← Ext.comp_assoc_of_second_deg_zero,
    (x.comp (Ext.mk₀ h.i) (add_zero i)).eq_zero_of_hasInjectiveDimensionLT n hi, Ext.zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y : C
h : Retract X Y
n : ℕ
inst_2 : HasInjectiveDimensionLT.{v, u} Y n
this : HasExt C := ⋯
i : ℕ
hi : n ≤ i
T : C
x : Ext T X i
rw : x.comp (Ext.mk₀ (𝟙 X)) ⋯ = 0
rw₁ : x.comp (Ext.mk₀ (h.i ≫ h.r)) ⋯ = 0
rw₂ : x.comp (Ext.comp (Ext.mk₀ h.i) (Ext.mk₀ h.r) ⋯) ⋯ = 0
rw₃ : (x.comp (Ext.mk₀ h.i) ⋯).comp (Ext.mk₀ h.r) ⋯ = 0
rw₄ : Eq (α := Ext T X i) (Ext.comp 0 (Ext.mk₀ h.r) ⋯) 0
⊢ x = 0
have rw₅ : Eq (α := Ext T X i) 0 0
Tactic: rw [← x.comp_mk₀_id, ← h.retract, ← Ext.mk₀_comp_mk₀, ← Ext.comp_assoc_of_second_deg_zero,
    (x.comp (Ext.mk₀ h.i) (add_zero i)).eq_zero_of_hasInjectiveDimensionLT n hi, Ext.zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
S : ShortComplex.{v, u} C
hS : S.ShortExact
n : ℕ
h₁ : HasInjectiveDimensionLT.{v, u} S.X₁ n
h₃ : HasInjectiveDimensionLT.{v, u} S.X₃ n
this : HasExt C := ⋯
⊢ HasInjectiveDimensionLT.{v, u} S.X₂ n
have rw : ∀ (i : ℕ), n ≤ i → ∀ ⦃Y : C⦄ (e : Ext Y S.X₂ i), e = 0
Tactic: rw [hasInjectiveDimensionLT_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
S : ShortComplex.{v, u} C
hS : S.ShortExact
n : ℕ
h₁ : HasInjectiveDimensionLT.{v, u} S.X₁ n
h₃ : HasInjectiveDimensionLT.{v, u} S.X₃ n
this : HasExt C := ⋯
i : ℕ
hi : n ≤ i
Y : C
x₂ : Ext Y S.X₂ i
⊢ x₂ = 0
have obtain : ∀ (x₃ : Ext Y S.X₁ i), x₃.comp (Ext.mk₀ S.f) sorry = 0
Tactic: obtain ⟨x₃, rfl⟩ := Ext.covariant_sequence_exact₂ _ hS x₂
    (Ext.eq_zero_of_hasInjectiveDimensionLT _ n hi)
Cleared hypotheses: x₂
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
S : ShortComplex.{v, u} C
hS : S.ShortExact
n : ℕ
h₁ : HasInjectiveDimensionLT.{v, u} S.X₁ n
h₃ : HasInjectiveDimensionLT.{v, u} S.X₃ n
this : HasExt C := ⋯
i : ℕ
hi : n ≤ i
Y : C
x₃ : Ext Y S.X₁ i
⊢ x₃.comp (Ext.mk₀ S.f) ⋯ = 0
have rw : Eq (α := Ext Y S.X₂ i) (Ext.comp 0 (Ext.mk₀ S.f) sorry) 0
Tactic: rw [x₃.eq_zero_of_hasInjectiveDimensionLT n hi, Ext.zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
S : ShortComplex.{v, u} C
hS : S.ShortExact
n : ℕ
h₁ : HasInjectiveDimensionLT.{v, u} S.X₁ n
h₃ : HasInjectiveDimensionLT.{v, u} S.X₃ n
this : HasExt C := ⋯
i : ℕ
hi : n ≤ i
Y : C
x₃ : Ext Y S.X₁ i
rw : Eq (α := Ext Y S.X₂ i) (Ext.comp 0 (Ext.mk₀ S.f) ⋯) 0
⊢ x₃.comp (Ext.mk₀ S.f) ⋯ = 0
have rw₁ : Eq (α := Ext Y S.X₂ i) 0 0
Tactic: rw [x₃.eq_zero_of_hasInjectiveDimensionLT n hi, Ext.zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
S : ShortComplex.{v, u} C
hS : S.ShortExact
n : ℕ
h₁ : HasInjectiveDimensionLT.{v, u} S.X₃ n
h₂ : HasInjectiveDimensionLT.{v, u} S.X₂ (n + 1)
this : HasExt C := ⋯
⊢ HasInjectiveDimensionLT.{v, u} S.X₁ (n + 1)
have rw : ∀ (i : ℕ), n + 1 ≤ i → ∀ ⦃Y : C⦄ (e : Ext Y S.X₁ i), e = 0
Tactic: rw [hasInjectiveDimensionLT_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
S : ShortComplex.{v, u} C
hS : S.ShortExact
n : ℕ
h₁ : HasInjectiveDimensionLT.{v, u} S.X₃ n
h₂ : HasInjectiveDimensionLT.{v, u} S.X₂ (n + 1)
this : HasExt C := ⋯
i : ℕ
hi : n + 1 ≤ i + 1
Y : C
x₃ : Ext Y S.X₁ (i + 1)
⊢ x₃ = 0
have succ : ∀ (x₁ : Ext Y S.X₃ i), x₁.comp hS.extClass sorry = 0
Tactic: obtain ⟨x₁, rfl⟩ := Ext.covariant_sequence_exact₁ _ hS x₃
      (Ext.eq_zero_of_hasInjectiveDimensionLT _ (n + 1) hi) rfl
Cleared hypotheses: x₃
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
S : ShortComplex.{v, u} C
hS : S.ShortExact
n : ℕ
h₁ : HasInjectiveDimensionLT.{v, u} S.X₃ n
h₂ : HasInjectiveDimensionLT.{v, u} S.X₂ (n + 1)
this : HasExt C := ⋯
i : ℕ
hi : n + 1 ≤ i + 1
Y : C
x₁ : Ext Y S.X₃ i
⊢ x₁.comp hS.extClass ⋯ = 0
have succ : Eq (α := Ext Y S.X₁ (i + 1)) (Ext.comp 0 hS.extClass sorry) 0
Tactic: rw [x₁.eq_zero_of_hasInjectiveDimensionLT n (by lia), Ext.zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
S : ShortComplex.{v, u} C
hS : S.ShortExact
n : ℕ
h₁ : HasInjectiveDimensionLT.{v, u} S.X₃ n
h₂ : HasInjectiveDimensionLT.{v, u} S.X₂ (n + 1)
this : HasExt C := ⋯
i : ℕ
hi : n + 1 ≤ i + 1
Y : C
x₁ : Ext Y S.X₃ i
succ : Eq (α := Ext Y S.X₁ (i + 1)) (Ext.comp 0 hS.extClass ⋯) 0
⊢ x₁.comp hS.extClass ⋯ = 0
have succ₁ : Eq (α := Ext Y S.X₁ (i + 1)) 0 0
Tactic: rw [x₁.eq_zero_of_hasInjectiveDimensionLT n (by lia), Ext.zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
S : ShortComplex.{v, u} C
hS : S.ShortExact
n : ℕ
h₂ : HasInjectiveDimensionLT.{v, u} S.X₂ n
h₃ : HasInjectiveDimensionLT.{v, u} S.X₁ (n + 1)
this : HasExt C := ⋯
⊢ HasInjectiveDimensionLT.{v, u} S.X₃ n
have rw : ∀ (i : ℕ), n ≤ i → ∀ ⦃Y : C⦄ (e : Ext Y S.X₃ i), e = 0
Tactic: rw [hasInjectiveDimensionLT_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
S : ShortComplex.{v, u} C
hS : S.ShortExact
n : ℕ
h₂ : HasInjectiveDimensionLT.{v, u} S.X₂ n
h₃ : HasInjectiveDimensionLT.{v, u} S.X₁ (n + 1)
this : HasExt C := ⋯
i : ℕ
hi : n ≤ i
Y : C
x₁ : Ext Y S.X₃ i
⊢ x₁ = 0
have obtain : ∀ (x₂ : Ext Y S.X₂ i), x₂.comp (Ext.mk₀ S.g) sorry = 0
Tactic: obtain ⟨x₂, rfl⟩ := Ext.covariant_sequence_exact₃ _ hS x₁ (add_comm _ _)
    (Ext.eq_zero_of_hasInjectiveDimensionLT _ (n + 1) (by lia))
Cleared hypotheses: x₁
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
S : ShortComplex.{v, u} C
hS : S.ShortExact
n : ℕ
h₂ : HasInjectiveDimensionLT.{v, u} S.X₂ n
h₃ : HasInjectiveDimensionLT.{v, u} S.X₁ (n + 1)
this : HasExt C := ⋯
i : ℕ
hi : n ≤ i
Y : C
x₂ : Ext Y S.X₂ i
⊢ x₂.comp (Ext.mk₀ S.g) ⋯ = 0
have rw : Eq (α := Ext Y S.X₃ i) (Ext.comp 0 (Ext.mk₀ S.g) sorry) 0
Tactic: rw [x₂.eq_zero_of_hasInjectiveDimensionLT n (by lia), Ext.zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
S : ShortComplex.{v, u} C
hS : S.ShortExact
n : ℕ
h₂ : HasInjectiveDimensionLT.{v, u} S.X₂ n
h₃ : HasInjectiveDimensionLT.{v, u} S.X₁ (n + 1)
this : HasExt C := ⋯
i : ℕ
hi : n ≤ i
Y : C
x₂ : Ext Y S.X₂ i
rw : Eq (α := Ext Y S.X₃ i) (Ext.comp 0 (Ext.mk₀ S.g) ⋯) 0
⊢ x₂.comp (Ext.mk₀ S.g) ⋯ = 0
have rw₁ : Eq (α := Ext Y S.X₃ i) 0 0
Tactic: rw [x₂.eq_zero_of_hasInjectiveDimensionLT n (by lia), Ext.zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
inst_3 : EnoughProjectives.{v, u} C
X : C
n : ℕ
hX : ∀ (Y : C), Subsingleton.{w + 1} (Ext.{w, v, u} Y X n)
this : ∀ ⦃d : ℕ⦄ ⦃Y : C⦄ (e : Ext.{w, v, u} Y X d) (k : ℕ), d = n + k → e = 0
i : ℕ
hi : n ≤ i
Y : C
e : Ext.{w, v, u} Y X i
⊢ e = 0
have obtain : ∀ (k : ℕ) (hi : n ≤ n + k) (e : Ext.{w, v, u} Y X (n + k)), e = 0
Tactic: obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le hi
Cleared hypotheses: i, hi, e
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
inst_3 : EnoughProjectives.{v, u} C
X : C
n : ℕ
hX : ∀ (Y : C), Subsingleton.{w + 1} (Ext.{w, v, u} Y X n)
⊢ HasInjectiveDimensionLT.{v, u} X n
have suffices : ∀ ⦃d : ℕ⦄ ⦃Y : C⦄ (e : Ext.{w, v, u} Y X d) (k : ℕ), d = n + k → e = 0
Tactic: suffices ∀ ⦃d : ℕ⦄ ⦃Y : C⦄ (e : Ext Y X d) (k : ℕ), d = n + k → e = 0 from
    HasInjectiveDimensionLT.mk (fun i hi Y e ↦ by
      obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le hi
      exact this e k rfl)
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
inst_3 : EnoughProjectives.{v, u} C
X : C
n : ℕ
hX : ∀ (Y : C), Subsingleton.{w + 1} (Ext.{w, v, u} Y X n)
d : ℕ
Y : C
e : Ext.{w, v, u} Y X d
hd : d = n + 0
⊢ e = 0
have hX₁ : ∀ (Y : C), Subsingleton.{w + 1} (Ext.{w, v, u} Y X d)
Tactic: obtain rfl : d = n := by simpa using hd
Cleared hypotheses: n, hX, hd
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Dimension (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : HasExt.{w, v, u} C
inst_3 : EnoughProjectives.{v, u} C
X : C
n : ℕ
hX : ∀ (Y : C), Subsingleton.{w + 1} (Ext.{w, v, u} Y X n)
d : ℕ
Y : C
e : Ext.{w, v, u} Y X d
hd : d = n + 0
hX₁ : ∀ (Y : C), Subsingleton.{w + 1} (Ext.{w, v, u} Y X d)
⊢ e = 0
have hd₁ : d = d + 0
Tactic: obtain rfl : d = n := by simpa using hd
Cleared hypotheses: n, hX, hd