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
        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
        apply succ_nsmul
      sub_eq_add_neg := 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 b : A₁ ⟶ A₂), a - b = a + -b

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
        apply Algebra.Hom.ext
        apply succ_nsmul
      sub_eq_add_neg := 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✝ b✝ : A₁ ⟶ A₂
⊢ a✝ - b✝ = a✝ + -b✝

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
        apply succ_nsmul
      sub_eq_add_neg := 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✝ : A₁ ⟶ A₂
⊢ (a✝ - b✝).f = (a✝ + -b✝).f

Next tactic:
apply sub_eq_add_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 simp only [Functor.map_sub, sub_comp, Endofunctor.Algebra.Hom.h, comp_sub] }
      zsmul := fun r α =>
        { f := r • 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
r : ℤ
α : A₁ ⟶ A₂
⊢ F.map (r • α.f) ≫ A₂.str = A₁.str ≫ (r • α.f)

Next tactic:
rw [comp_zsmul, Functor.map_zsmul, zsmul_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
        intros
        apply Algebra.Hom.ext
        apply zero_smul
      nsmul_succ := by
        intros
        apply Algebra.Hom.ext
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Algebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_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), (fun r α => Algebra.Hom.mk (r • α.f)) 0 a = 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
        apply Algebra.Hom.ext
        apply zero_smul
      nsmul_succ := by
        intros
        apply Algebra.Hom.ext
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Algebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_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
⊢ (fun r α => Algebra.Hom.mk (r • α.f)) 0 a✝ = 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
        apply zero_smul
      nsmul_succ := by
        intros
        apply Algebra.Hom.ext
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Algebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_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
⊢ ((fun r α => Algebra.Hom.mk (r • α.f)) 0 a✝).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
        intros
        apply Algebra.Hom.ext
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Algebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_zero' := by
        intros
        apply Algebra.Hom.ext
        apply zero_smul
      zsmul_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) (a : A₁ ⟶ A₂),
    (fun r α => Algebra.Hom.mk (r • α.f)) (Int.ofNat (Nat.succ n)) a =
      a + (fun r α => Algebra.Hom.mk (r • α.f)) (Int.ofNat n) 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
        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
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Algebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_zero' := by
        intros
        apply Algebra.Hom.ext
        apply zero_smul
      zsmul_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
a✝ : A₁ ⟶ A₂
⊢ (fun r α => Algebra.Hom.mk (r • α.f)) (Int.ofNat (Nat.succ n✝)) a✝ =
    a✝ + (fun r α => Algebra.Hom.mk (r • α.f)) (Int.ofNat n✝) 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
        apply add_zero
      nsmul_zero := by
        intros
        apply Algebra.Hom.ext
        apply zero_smul
      nsmul_succ := by
        intros
        apply Algebra.Hom.ext
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Algebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_zero' := by
        intros
        apply Algebra.Hom.ext
        apply zero_smul
      zsmul_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
a✝ : A₁ ⟶ A₂
⊢ ((fun r α => Algebra.Hom.mk (r • α.f)) (Int.ofNat (Nat.succ n✝)) a✝).f =
    (a✝ + (fun r α => Algebra.Hom.mk (r • α.f)) (Int.ofNat n✝) a✝).f

Next tactic:
dsimp
/-- 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
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Algebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_zero' := by
        intros
        apply Algebra.Hom.ext
        apply zero_smul
      zsmul_succ' := by
        intros
        apply Algebra.Hom.ext
        dsimp
        

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
a✝ : A₁ ⟶ A₂
⊢ ↑(Nat.succ n✝) • a✝.f = (a✝ + Algebra.Hom.mk (↑n✝ • a✝.f)).f

Next tactic:
simp only [coe_nat_zsmul, 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 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
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Algebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_zero' := by
        intros
        apply Algebra.Hom.ext
        apply zero_smul
      zsmul_succ' := by
        intros
        apply Algebra.Hom.ext
        dsimp
        simp only [coe_nat_zsmul, succ_nsmul]
        

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
a✝ : A₁ ⟶ A₂
⊢ a✝.f + n✝ • a✝.f = (a✝ + Algebra.Hom.mk (n✝ • a✝.f)).f

Next tactic:
rfl
/-- 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
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Algebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_zero' := by
        intros
        apply Algebra.Hom.ext
        apply zero_smul
      zsmul_succ' := by
        intros
        apply Algebra.Hom.ext
        dsimp
        simp only [coe_nat_zsmul, succ_nsmul]
        rfl
      zsmul_neg' := 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) (a : A₁ ⟶ A₂),
    (fun r α => Algebra.Hom.mk (r • α.f)) (Int.negSucc n) a = -(fun r α => Algebra.Hom.mk (r • α.f)) (↑(Nat.succ n)) a

Next tactic:
intros