l3lab/ntp-mathlib huggingface dataset:
@[simps]
instance Endofunctor.coalgebraPreadditive : Preadditive (Endofunctor.Coalgebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, comp_add, Endofunctor.Coalgebra.Hom.h, add_comp] }
      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 [Functor.map_nsmul, comp_nsmul, Endofunctor.Coalgebra.Hom.h, nsmul_comp] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, comp_neg, Endofunctor.Coalgebra.Hom.h, neg_comp] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, comp_sub, Endofunctor.Coalgebra.Hom.h, sub_comp] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [Functor.map_zsmul, comp_zsmul, Endofunctor.Coalgebra.Hom.h, zsmul_comp] }
      add_assoc := by
        intros
        apply Coalgebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply add_zero
      nsmul_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      nsmul_succ := by
        intros
        apply Coalgebra.Hom.ext
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Coalgebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_zero' := by
        intros
        apply Coalgebra.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 : Coalgebra F
⊢ ∀ (n : ℕ) (a : A₁ ⟶ A₂),
    (fun r α => Coalgebra.Hom.mk (r • α.f)) (Int.ofNat (Nat.succ n)) a =
      a + (fun r α => Coalgebra.Hom.mk (r • α.f)) (Int.ofNat n) a

Next tactic:
intros
@[simps]
instance Endofunctor.coalgebraPreadditive : Preadditive (Endofunctor.Coalgebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, comp_add, Endofunctor.Coalgebra.Hom.h, add_comp] }
      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 [Functor.map_nsmul, comp_nsmul, Endofunctor.Coalgebra.Hom.h, nsmul_comp] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, comp_neg, Endofunctor.Coalgebra.Hom.h, neg_comp] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, comp_sub, Endofunctor.Coalgebra.Hom.h, sub_comp] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [Functor.map_zsmul, comp_zsmul, Endofunctor.Coalgebra.Hom.h, zsmul_comp] }
      add_assoc := by
        intros
        apply Coalgebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply add_zero
      nsmul_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      nsmul_succ := by
        intros
        apply Coalgebra.Hom.ext
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Coalgebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_zero' := by
        intros
        apply Coalgebra.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 : Coalgebra F
n✝ : ℕ
a✝ : A₁ ⟶ A₂
⊢ (fun r α => Coalgebra.Hom.mk (r • α.f)) (Int.ofNat (Nat.succ n✝)) a✝ =
    a✝ + (fun r α => Coalgebra.Hom.mk (r • α.f)) (Int.ofNat n✝) a✝

Next tactic:
apply Coalgebra.Hom.ext
@[simps]
instance Endofunctor.coalgebraPreadditive : Preadditive (Endofunctor.Coalgebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, comp_add, Endofunctor.Coalgebra.Hom.h, add_comp] }
      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 [Functor.map_nsmul, comp_nsmul, Endofunctor.Coalgebra.Hom.h, nsmul_comp] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, comp_neg, Endofunctor.Coalgebra.Hom.h, neg_comp] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, comp_sub, Endofunctor.Coalgebra.Hom.h, sub_comp] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [Functor.map_zsmul, comp_zsmul, Endofunctor.Coalgebra.Hom.h, zsmul_comp] }
      add_assoc := by
        intros
        apply Coalgebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply add_zero
      nsmul_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      nsmul_succ := by
        intros
        apply Coalgebra.Hom.ext
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Coalgebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_zero' := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      zsmul_succ' := by
        intros
        apply Coalgebra.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 : Coalgebra F
n✝ : ℕ
a✝ : A₁ ⟶ A₂
⊢ ((fun r α => Coalgebra.Hom.mk (r • α.f)) (Int.ofNat (Nat.succ n✝)) a✝).f =
    (a✝ + (fun r α => Coalgebra.Hom.mk (r • α.f)) (Int.ofNat n✝) a✝).f

Next tactic:
dsimp
@[simps]
instance Endofunctor.coalgebraPreadditive : Preadditive (Endofunctor.Coalgebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, comp_add, Endofunctor.Coalgebra.Hom.h, add_comp] }
      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 [Functor.map_nsmul, comp_nsmul, Endofunctor.Coalgebra.Hom.h, nsmul_comp] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, comp_neg, Endofunctor.Coalgebra.Hom.h, neg_comp] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, comp_sub, Endofunctor.Coalgebra.Hom.h, sub_comp] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [Functor.map_zsmul, comp_zsmul, Endofunctor.Coalgebra.Hom.h, zsmul_comp] }
      add_assoc := by
        intros
        apply Coalgebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply add_zero
      nsmul_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      nsmul_succ := by
        intros
        apply Coalgebra.Hom.ext
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Coalgebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_zero' := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      zsmul_succ' := by
        intros
        apply Coalgebra.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 : Coalgebra F
n✝ : ℕ
a✝ : A₁ ⟶ A₂
⊢ ↑(Nat.succ n✝) • a✝.f = (a✝ + Coalgebra.Hom.mk (↑n✝ • a✝.f)).f

Next tactic:
simp only [coe_nat_zsmul, succ_nsmul]
@[simps]
instance Endofunctor.coalgebraPreadditive : Preadditive (Endofunctor.Coalgebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, comp_add, Endofunctor.Coalgebra.Hom.h, add_comp] }
      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 [Functor.map_nsmul, comp_nsmul, Endofunctor.Coalgebra.Hom.h, nsmul_comp] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, comp_neg, Endofunctor.Coalgebra.Hom.h, neg_comp] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, comp_sub, Endofunctor.Coalgebra.Hom.h, sub_comp] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [Functor.map_zsmul, comp_zsmul, Endofunctor.Coalgebra.Hom.h, zsmul_comp] }
      add_assoc := by
        intros
        apply Coalgebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply add_zero
      nsmul_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      nsmul_succ := by
        intros
        apply Coalgebra.Hom.ext
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Coalgebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_zero' := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      zsmul_succ' := by
        intros
        apply Coalgebra.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 : Coalgebra F
n✝ : ℕ
a✝ : A₁ ⟶ A₂
⊢ a✝.f + n✝ • a✝.f = (a✝ + Coalgebra.Hom.mk (n✝ • a✝.f)).f

Next tactic:
rfl
@[simps]
instance Endofunctor.coalgebraPreadditive : Preadditive (Endofunctor.Coalgebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, comp_add, Endofunctor.Coalgebra.Hom.h, add_comp] }
      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 [Functor.map_nsmul, comp_nsmul, Endofunctor.Coalgebra.Hom.h, nsmul_comp] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, comp_neg, Endofunctor.Coalgebra.Hom.h, neg_comp] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, comp_sub, Endofunctor.Coalgebra.Hom.h, sub_comp] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [Functor.map_zsmul, comp_zsmul, Endofunctor.Coalgebra.Hom.h, zsmul_comp] }
      add_assoc := by
        intros
        apply Coalgebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply add_zero
      nsmul_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      nsmul_succ := by
        intros
        apply Coalgebra.Hom.ext
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Coalgebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_zero' := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      zsmul_succ' := by
        intros
        apply Coalgebra.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 : Coalgebra F
⊢ ∀ (n : ℕ) (a : A₁ ⟶ A₂),
    (fun r α => Coalgebra.Hom.mk (r • α.f)) (Int.negSucc n) a =
      -(fun r α => Coalgebra.Hom.mk (r • α.f)) (↑(Nat.succ n)) a

Next tactic:
intros
@[simps]
instance Endofunctor.coalgebraPreadditive : Preadditive (Endofunctor.Coalgebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, comp_add, Endofunctor.Coalgebra.Hom.h, add_comp] }
      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 [Functor.map_nsmul, comp_nsmul, Endofunctor.Coalgebra.Hom.h, nsmul_comp] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, comp_neg, Endofunctor.Coalgebra.Hom.h, neg_comp] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, comp_sub, Endofunctor.Coalgebra.Hom.h, sub_comp] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [Functor.map_zsmul, comp_zsmul, Endofunctor.Coalgebra.Hom.h, zsmul_comp] }
      add_assoc := by
        intros
        apply Coalgebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply add_zero
      nsmul_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      nsmul_succ := by
        intros
        apply Coalgebra.Hom.ext
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Coalgebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_zero' := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      zsmul_succ' := by
        intros
        apply Coalgebra.Hom.ext
        dsimp
        simp only [coe_nat_zsmul, succ_nsmul]
        rfl
      zsmul_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 : Coalgebra F
n✝ : ℕ
a✝ : A₁ ⟶ A₂
⊢ (fun r α => Coalgebra.Hom.mk (r • α.f)) (Int.negSucc n✝) a✝ =
    -(fun r α => Coalgebra.Hom.mk (r • α.f)) (↑(Nat.succ n✝)) a✝

Next tactic:
apply Coalgebra.Hom.ext
@[simps]
instance Endofunctor.coalgebraPreadditive : Preadditive (Endofunctor.Coalgebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, comp_add, Endofunctor.Coalgebra.Hom.h, add_comp] }
      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 [Functor.map_nsmul, comp_nsmul, Endofunctor.Coalgebra.Hom.h, nsmul_comp] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, comp_neg, Endofunctor.Coalgebra.Hom.h, neg_comp] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, comp_sub, Endofunctor.Coalgebra.Hom.h, sub_comp] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [Functor.map_zsmul, comp_zsmul, Endofunctor.Coalgebra.Hom.h, zsmul_comp] }
      add_assoc := by
        intros
        apply Coalgebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply add_zero
      nsmul_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      nsmul_succ := by
        intros
        apply Coalgebra.Hom.ext
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Coalgebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_zero' := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      zsmul_succ' := by
        intros
        apply Coalgebra.Hom.ext
        dsimp
        simp only [coe_nat_zsmul, succ_nsmul]
        rfl
      zsmul_neg' := by
        intros
        apply Coalgebra.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 : Coalgebra F
n✝ : ℕ
a✝ : A₁ ⟶ A₂
⊢ ((fun r α => Coalgebra.Hom.mk (r • α.f)) (Int.negSucc n✝) a✝).f =
    (-(fun r α => Coalgebra.Hom.mk (r • α.f)) (↑(Nat.succ n✝)) a✝).f

Next tactic:
simp only [negSucc_zsmul, neg_inj, nsmul_eq_smul_cast ℤ]
@[simps]
instance Endofunctor.coalgebraPreadditive : Preadditive (Endofunctor.Coalgebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, comp_add, Endofunctor.Coalgebra.Hom.h, add_comp] }
      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 [Functor.map_nsmul, comp_nsmul, Endofunctor.Coalgebra.Hom.h, nsmul_comp] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, comp_neg, Endofunctor.Coalgebra.Hom.h, neg_comp] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, comp_sub, Endofunctor.Coalgebra.Hom.h, sub_comp] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [Functor.map_zsmul, comp_zsmul, Endofunctor.Coalgebra.Hom.h, zsmul_comp] }
      add_assoc := by
        intros
        apply Coalgebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply add_zero
      nsmul_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      nsmul_succ := by
        intros
        apply Coalgebra.Hom.ext
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Coalgebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_zero' := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      zsmul_succ' := by
        intros
        apply Coalgebra.Hom.ext
        dsimp
        simp only [coe_nat_zsmul, succ_nsmul]
        rfl
      zsmul_neg' := by
        intros
        apply Coalgebra.Hom.ext
        simp only [negSucc_zsmul, neg_inj, nsmul_eq_smul_cast ℤ]
      add_left_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 : Coalgebra F
⊢ ∀ (a : A₁ ⟶ A₂), -a + a = 0

Next tactic:
intros
@[simps]
instance Endofunctor.coalgebraPreadditive : Preadditive (Endofunctor.Coalgebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, comp_add, Endofunctor.Coalgebra.Hom.h, add_comp] }
      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 [Functor.map_nsmul, comp_nsmul, Endofunctor.Coalgebra.Hom.h, nsmul_comp] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, comp_neg, Endofunctor.Coalgebra.Hom.h, neg_comp] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, comp_sub, Endofunctor.Coalgebra.Hom.h, sub_comp] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [Functor.map_zsmul, comp_zsmul, Endofunctor.Coalgebra.Hom.h, zsmul_comp] }
      add_assoc := by
        intros
        apply Coalgebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply add_zero
      nsmul_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      nsmul_succ := by
        intros
        apply Coalgebra.Hom.ext
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Coalgebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_zero' := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      zsmul_succ' := by
        intros
        apply Coalgebra.Hom.ext
        dsimp
        simp only [coe_nat_zsmul, succ_nsmul]
        rfl
      zsmul_neg' := by
        intros
        apply Coalgebra.Hom.ext
        simp only [negSucc_zsmul, neg_inj, nsmul_eq_smul_cast ℤ]
      add_left_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 : Coalgebra F
a✝ : A₁ ⟶ A₂
⊢ -a✝ + a✝ = 0

Next tactic:
apply Coalgebra.Hom.ext
@[simps]
instance Endofunctor.coalgebraPreadditive : Preadditive (Endofunctor.Coalgebra F) where
  homGroup A₁ A₂ :=
    { add := fun α β =>
        { f := α.f + β.f
          h := by simp only [Functor.map_add, comp_add, Endofunctor.Coalgebra.Hom.h, add_comp] }
      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 [Functor.map_nsmul, comp_nsmul, Endofunctor.Coalgebra.Hom.h, nsmul_comp] }
      neg := fun α =>
        { f := -α.f
          h := by simp only [Functor.map_neg, comp_neg, Endofunctor.Coalgebra.Hom.h, neg_comp] }
      sub := fun α β =>
        { f := α.f - β.f
          h := by simp only [Functor.map_sub, comp_sub, Endofunctor.Coalgebra.Hom.h, sub_comp] }
      zsmul := fun r α =>
        { f := r • α.f
          h := by rw [Functor.map_zsmul, comp_zsmul, Endofunctor.Coalgebra.Hom.h, zsmul_comp] }
      add_assoc := by
        intros
        apply Coalgebra.Hom.ext
        apply add_assoc
      zero_add := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_add
      add_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply add_zero
      nsmul_zero := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      nsmul_succ := by
        intros
        apply Coalgebra.Hom.ext
        apply succ_nsmul
      sub_eq_add_neg := by
        intros
        apply Coalgebra.Hom.ext
        apply sub_eq_add_neg
      zsmul_zero' := by
        intros
        apply Coalgebra.Hom.ext
        apply zero_smul
      zsmul_succ' := by
        intros
        apply Coalgebra.Hom.ext
        dsimp
        simp only [coe_nat_zsmul, succ_nsmul]
        rfl
      zsmul_neg' := by
        intros
        apply Coalgebra.Hom.ext
        simp only [negSucc_zsmul, neg_inj, nsmul_eq_smul_cast ℤ]
      add_left_neg := by
        intros
        apply Coalgebra.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 : Coalgebra F
a✝ : A₁ ⟶ A₂
⊢ (-a✝ + a✝).f = 0.f

Next tactic:
apply add_left_neg