l3lab/ntp-mathlib huggingface dataset:
/-- The category of algebras over an additive endofunctor on a preadditive category is preadditive.
-/
@[simps]
instance Endofunctor.algebraPreadditive : Preadditive (Endofunctor.Algebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add] }
      zero :=
        { f := 0
          h := by simp only [Functor.map_zero, zero_comp, comp_zero] }
      nsmul := fun n α =>
        { f := n • α.f
          h := by rw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, neg_comp, Endofunctor.Algebra.Hom.h, comp_neg] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, sub_comp, Endofunctor.Algebra.Hom.h, comp_sub] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [comp_zsmul, Functor.map_zsmul, zsmul_comp, Endofunctor.Algebra.Hom.h] }
      add_assoc := by
        intros
        apply Algebra.Hom.ext
        

Proof state:
case f
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : Functor.Additive F
A₁ A₂ : Algebra F
a✝ b✝ c✝ : A₁ ⟶ A₂
⊢ (a✝ + b✝ + c✝).f = (a✝ + (b✝ + c✝)).f

Next tactic:
apply add_assoc
/-- The category of algebras over an additive endofunctor on a preadditive category is preadditive.
-/
@[simps]
instance Endofunctor.algebraPreadditive : Preadditive (Endofunctor.Algebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add] }
      zero :=
        { f := 0
          h := by 

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : Functor.Additive F
A₁ A₂ : Algebra F
⊢ F.map 0 ≫ A₂.str = A₁.str ≫ 0

Next tactic:
simp only [Functor.map_zero, zero_comp, comp_zero]
/-- The category of algebras over an additive endofunctor on a preadditive category is preadditive.
-/
@[simps]
instance Endofunctor.algebraPreadditive : Preadditive (Endofunctor.Algebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add] }
      zero :=
        { f := 0
          h := by simp only [Functor.map_zero, zero_comp, comp_zero] }
      nsmul := fun n α =>
        { f := n • α.f
          h := by rw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, neg_comp, Endofunctor.Algebra.Hom.h, comp_neg] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, sub_comp, Endofunctor.Algebra.Hom.h, comp_sub] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [comp_zsmul, Functor.map_zsmul, zsmul_comp, Endofunctor.Algebra.Hom.h] }
      add_assoc := by
        intros
        apply Algebra.Hom.ext
        apply add_assoc
      zero_add := by
        

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : Functor.Additive F
A₁ A₂ : Algebra F
⊢ ∀ (a : A₁ ⟶ A₂), 0 + a = a

Next tactic:
intros
/-- The category of algebras over an additive endofunctor on a preadditive category is preadditive.
-/
@[simps]
instance Endofunctor.algebraPreadditive : Preadditive (Endofunctor.Algebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add] }
      zero :=
        { f := 0
          h := by simp only [Functor.map_zero, zero_comp, comp_zero] }
      nsmul := fun n α =>
        { f := n • α.f
          h := by rw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, neg_comp, Endofunctor.Algebra.Hom.h, comp_neg] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, sub_comp, Endofunctor.Algebra.Hom.h, comp_sub] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [comp_zsmul, Functor.map_zsmul, zsmul_comp, Endofunctor.Algebra.Hom.h] }
      add_assoc := by
        intros
        apply Algebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : Functor.Additive F
A₁ A₂ : Algebra F
a✝ : A₁ ⟶ A₂
⊢ 0 + a✝ = a✝

Next tactic:
apply Algebra.Hom.ext
/-- The category of algebras over an additive endofunctor on a preadditive category is preadditive.
-/
@[simps]
instance Endofunctor.algebraPreadditive : Preadditive (Endofunctor.Algebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add] }
      zero :=
        { f := 0
          h := by simp only [Functor.map_zero, zero_comp, comp_zero] }
      nsmul := fun n α =>
        { f := n • α.f
          h := by rw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, neg_comp, Endofunctor.Algebra.Hom.h, comp_neg] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, sub_comp, Endofunctor.Algebra.Hom.h, comp_sub] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [comp_zsmul, Functor.map_zsmul, zsmul_comp, Endofunctor.Algebra.Hom.h] }
      add_assoc := by
        intros
        apply Algebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Algebra.Hom.ext
        

Proof state:
case f
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : Functor.Additive F
A₁ A₂ : Algebra F
a✝ : A₁ ⟶ A₂
⊢ (0 + a✝).f = a✝.f

Next tactic:
apply zero_add
/-- The category of algebras over an additive endofunctor on a preadditive category is preadditive.
-/
@[simps]
instance Endofunctor.algebraPreadditive : Preadditive (Endofunctor.Algebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add] }
      zero :=
        { f := 0
          h := by simp only [Functor.map_zero, zero_comp, comp_zero] }
      nsmul := fun n α =>
        { f := n • α.f
          h := by rw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, neg_comp, Endofunctor.Algebra.Hom.h, comp_neg] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, sub_comp, Endofunctor.Algebra.Hom.h, comp_sub] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [comp_zsmul, Functor.map_zsmul, zsmul_comp, Endofunctor.Algebra.Hom.h] }
      add_assoc := by
        intros
        apply Algebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Algebra.Hom.ext
        apply zero_add
      add_zero := by
        

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : Functor.Additive F
A₁ A₂ : Algebra F
⊢ ∀ (a : A₁ ⟶ A₂), a + 0 = a

Next tactic:
intros
/-- The category of algebras over an additive endofunctor on a preadditive category is preadditive.
-/
@[simps]
instance Endofunctor.algebraPreadditive : Preadditive (Endofunctor.Algebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add] }
      zero :=
        { f := 0
          h := by simp only [Functor.map_zero, zero_comp, comp_zero] }
      nsmul := fun n α =>
        { f := n • α.f
          h := by rw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, neg_comp, Endofunctor.Algebra.Hom.h, comp_neg] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, sub_comp, Endofunctor.Algebra.Hom.h, comp_sub] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [comp_zsmul, Functor.map_zsmul, zsmul_comp, Endofunctor.Algebra.Hom.h] }
      add_assoc := by
        intros
        apply Algebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Algebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : Functor.Additive F
A₁ A₂ : Algebra F
a✝ : A₁ ⟶ A₂
⊢ a✝ + 0 = a✝

Next tactic:
apply Algebra.Hom.ext
/-- The category of algebras over an additive endofunctor on a preadditive category is preadditive.
-/
@[simps]
instance Endofunctor.algebraPreadditive : Preadditive (Endofunctor.Algebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add] }
      zero :=
        { f := 0
          h := by simp only [Functor.map_zero, zero_comp, comp_zero] }
      nsmul := fun n α =>
        { f := n • α.f
          h := by rw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, neg_comp, Endofunctor.Algebra.Hom.h, comp_neg] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, sub_comp, Endofunctor.Algebra.Hom.h, comp_sub] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [comp_zsmul, Functor.map_zsmul, zsmul_comp, Endofunctor.Algebra.Hom.h] }
      add_assoc := by
        intros
        apply Algebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Algebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Algebra.Hom.ext
        

Proof state:
case f
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : Functor.Additive F
A₁ A₂ : Algebra F
a✝ : A₁ ⟶ A₂
⊢ (a✝ + 0).f = a✝.f

Next tactic:
apply add_zero
/-- The category of algebras over an additive endofunctor on a preadditive category is preadditive.
-/
@[simps]
instance Endofunctor.algebraPreadditive : Preadditive (Endofunctor.Algebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add] }
      zero :=
        { f := 0
          h := by simp only [Functor.map_zero, zero_comp, comp_zero] }
      nsmul := fun n α =>
        { f := n • α.f
          h := by 

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : Functor.Additive F
A₁ A₂ : Algebra F
n : ℕ
α : A₁ ⟶ A₂
⊢ F.map (n • α.f) ≫ A₂.str = A₁.str ≫ (n • α.f)

Next tactic:
rw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h]
/-- The category of algebras over an additive endofunctor on a preadditive category is preadditive.
-/
@[simps]
instance Endofunctor.algebraPreadditive : Preadditive (Endofunctor.Algebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add] }
      zero :=
        { f := 0
          h := by simp only [Functor.map_zero, zero_comp, comp_zero] }
      nsmul := fun n α =>
        { f := n • α.f
          h := by rw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, neg_comp, Endofunctor.Algebra.Hom.h, comp_neg] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, sub_comp, Endofunctor.Algebra.Hom.h, comp_sub] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [comp_zsmul, Functor.map_zsmul, zsmul_comp, Endofunctor.Algebra.Hom.h] }
      add_assoc := by
        intros
        apply Algebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Algebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Algebra.Hom.ext
        apply add_zero
      nsmul_zero := by
        

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : Functor.Additive F
A₁ A₂ : Algebra F
⊢ ∀ (x : A₁ ⟶ A₂), (fun n α => Algebra.Hom.mk (n • α.f)) 0 x = 0

Next tactic:
intros
/-- The category of algebras over an additive endofunctor on a preadditive category is preadditive.
-/
@[simps]
instance Endofunctor.algebraPreadditive : Preadditive (Endofunctor.Algebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add] }
      zero :=
        { f := 0
          h := by simp only [Functor.map_zero, zero_comp, comp_zero] }
      nsmul := fun n α =>
        { f := n • α.f
          h := by rw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, neg_comp, Endofunctor.Algebra.Hom.h, comp_neg] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, sub_comp, Endofunctor.Algebra.Hom.h, comp_sub] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [comp_zsmul, Functor.map_zsmul, zsmul_comp, Endofunctor.Algebra.Hom.h] }
      add_assoc := by
        intros
        apply Algebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Algebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Algebra.Hom.ext
        apply add_zero
      nsmul_zero := by
        intros
        

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : Functor.Additive F
A₁ A₂ : Algebra F
x✝ : A₁ ⟶ A₂
⊢ (fun n α => Algebra.Hom.mk (n • α.f)) 0 x✝ = 0

Next tactic:
apply Algebra.Hom.ext
/-- The category of algebras over an additive endofunctor on a preadditive category is preadditive.
-/
@[simps]
instance Endofunctor.algebraPreadditive : Preadditive (Endofunctor.Algebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add] }
      zero :=
        { f := 0
          h := by simp only [Functor.map_zero, zero_comp, comp_zero] }
      nsmul := fun n α =>
        { f := n • α.f
          h := by rw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, neg_comp, Endofunctor.Algebra.Hom.h, comp_neg] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, sub_comp, Endofunctor.Algebra.Hom.h, comp_sub] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [comp_zsmul, Functor.map_zsmul, zsmul_comp, Endofunctor.Algebra.Hom.h] }
      add_assoc := by
        intros
        apply Algebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Algebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Algebra.Hom.ext
        apply add_zero
      nsmul_zero := by
        intros
        apply Algebra.Hom.ext
        

Proof state:
case f
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : Functor.Additive F
A₁ A₂ : Algebra F
x✝ : A₁ ⟶ A₂
⊢ ((fun n α => Algebra.Hom.mk (n • α.f)) 0 x✝).f = 0.f

Next tactic:
apply zero_smul
/-- The category of algebras over an additive endofunctor on a preadditive category is preadditive.
-/
@[simps]
instance Endofunctor.algebraPreadditive : Preadditive (Endofunctor.Algebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add] }
      zero :=
        { f := 0
          h := by simp only [Functor.map_zero, zero_comp, comp_zero] }
      nsmul := fun n α =>
        { f := n • α.f
          h := by rw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, neg_comp, Endofunctor.Algebra.Hom.h, comp_neg] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, sub_comp, Endofunctor.Algebra.Hom.h, comp_sub] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [comp_zsmul, Functor.map_zsmul, zsmul_comp, Endofunctor.Algebra.Hom.h] }
      add_assoc := by
        intros
        apply Algebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Algebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Algebra.Hom.ext
        apply add_zero
      nsmul_zero := by
        intros
        apply Algebra.Hom.ext
        apply zero_smul
      nsmul_succ := by
        

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : Functor.Additive F
A₁ A₂ : Algebra F
⊢ ∀ (n : ℕ) (x : A₁ ⟶ A₂),
    (fun n α => Algebra.Hom.mk (n • α.f)) (n + 1) x = x + (fun n α => Algebra.Hom.mk (n • α.f)) n x

Next tactic:
intros
/-- The category of algebras over an additive endofunctor on a preadditive category is preadditive.
-/
@[simps]
instance Endofunctor.algebraPreadditive : Preadditive (Endofunctor.Algebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add] }
      zero :=
        { f := 0
          h := by simp only [Functor.map_zero, zero_comp, comp_zero] }
      nsmul := fun n α =>
        { f := n • α.f
          h := by rw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, neg_comp, Endofunctor.Algebra.Hom.h, comp_neg] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, sub_comp, Endofunctor.Algebra.Hom.h, comp_sub] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [comp_zsmul, Functor.map_zsmul, zsmul_comp, Endofunctor.Algebra.Hom.h] }
      add_assoc := by
        intros
        apply Algebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Algebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Algebra.Hom.ext
        apply add_zero
      nsmul_zero := by
        intros
        apply Algebra.Hom.ext
        apply zero_smul
      nsmul_succ := by
        intros
        

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : Functor.Additive F
A₁ A₂ : Algebra F
n✝ : ℕ
x✝ : A₁ ⟶ A₂
⊢ (fun n α => Algebra.Hom.mk (n • α.f)) (n✝ + 1) x✝ = x✝ + (fun n α => Algebra.Hom.mk (n • α.f)) n✝ x✝

Next tactic:
apply Algebra.Hom.ext
/-- The category of algebras over an additive endofunctor on a preadditive category is preadditive.
-/
@[simps]
instance Endofunctor.algebraPreadditive : Preadditive (Endofunctor.Algebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add] }
      zero :=
        { f := 0
          h := by simp only [Functor.map_zero, zero_comp, comp_zero] }
      nsmul := fun n α =>
        { f := n • α.f
          h := by rw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, neg_comp, Endofunctor.Algebra.Hom.h, comp_neg] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, sub_comp, Endofunctor.Algebra.Hom.h, comp_sub] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [comp_zsmul, Functor.map_zsmul, zsmul_comp, Endofunctor.Algebra.Hom.h] }
      add_assoc := by
        intros
        apply Algebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Algebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Algebra.Hom.ext
        apply add_zero
      nsmul_zero := by
        intros
        apply Algebra.Hom.ext
        apply zero_smul
      nsmul_succ := by
        intros
        apply Algebra.Hom.ext
        

Proof state:
case f
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : Functor.Additive F
A₁ A₂ : Algebra F
n✝ : ℕ
x✝ : A₁ ⟶ A₂
⊢ ((fun n α => Algebra.Hom.mk (n • α.f)) (n✝ + 1) x✝).f = (x✝ + (fun n α => Algebra.Hom.mk (n • α.f)) n✝ x✝).f

Next tactic:
apply succ_nsmul
/-- The category of algebras over an additive endofunctor on a preadditive category is preadditive.
-/
@[simps]
instance Endofunctor.algebraPreadditive : Preadditive (Endofunctor.Algebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add] }
      zero :=
        { f := 0
          h := by simp only [Functor.map_zero, zero_comp, comp_zero] }
      nsmul := fun n α =>
        { f := n • α.f
          h := by rw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h] }
      neg := fun α =>
        { f := -α.f
          h := by 

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : Functor.Additive F
A₁ A₂ : Algebra F
α : A₁ ⟶ A₂
⊢ F.map (-α.f) ≫ A₂.str = A₁.str ≫ (-α.f)

Next tactic:
simp only [Functor.map_neg, neg_comp, Endofunctor.Algebra.Hom.h, comp_neg]
/-- The category of algebras over an additive endofunctor on a preadditive category is preadditive.
-/
@[simps]
instance Endofunctor.algebraPreadditive : Preadditive (Endofunctor.Algebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, add_comp, Endofunctor.Algebra.Hom.h, comp_add] }
      zero :=
        { f := 0
          h := by simp only [Functor.map_zero, zero_comp, comp_zero] }
      nsmul := fun n α =>
        { f := n • α.f
          h := by rw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, neg_comp, Endofunctor.Algebra.Hom.h, comp_neg] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by 

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Preadditive C
F : C ⥤ C
inst✝ : Functor.Additive F
A₁ A₂ : Algebra F
α β : A₁ ⟶ A₂
⊢ F.map (α.f - β.f) ≫ A₂.str = A₁.str ≫ (α.f - β.f)

Next tactic:
simp only [Functor.map_sub, sub_comp, Endofunctor.Algebra.Hom.h, comp_sub]