l3lab/ntp-mathlib huggingface dataset:
/-- The function from the colimit commutative ring to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt := by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl => rfl
    | symm x y _ ih => exact ih.symm
    | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2
    | map j j' f x => exact RingHom.congr_fun (s.ι.naturality f) x
    | zero j => simp
    | one j => simp
    | neg j x => simp
    | add j x y => simp
    | mul j x y => simp
    | neg_1 x x' r ih => dsimp; rw [ih]
    | add_1 x x' y r ih => dsimp; rw [ih]
    | add_2 x y y' r ih => dsimp; rw [ih]
    | mul_1 x x' y r ih => dsimp; rw [ih]
    | mul_2 x y y' r ih => dsimp; rw [ih]
    | zero_add x => dsimp; rw [zero_add]
    | add_zero x => dsimp; rw [add_zero]
    | one_mul x => dsimp; rw [one_mul]
    | mul_one x => dsimp; rw [mul_one]
    | add_left_neg x => dsimp; rw [add_left_neg]
    | add_comm x y => dsimp; rw [add_comm]
    | mul_comm x y => dsimp; 

Proof state:
case a.mul_comm
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCat
s : Cocone F
x✝ y✝ x y : Prequotient F
⊢ descFunLift F s x * descFunLift F s y = descFunLift F s y * descFunLift F s x

Next tactic:
rw [mul_comm]
/-- The function from the colimit commutative ring to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt := by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl => rfl
    | symm x y _ ih => exact ih.symm
    | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2
    | map j j' f x => exact RingHom.congr_fun (s.ι.naturality f) x
    | zero j => simp
    | one j => simp
    | neg j x => simp
    | add j x y => simp
    | mul j x y => simp
    | neg_1 x x' r ih => dsimp; rw [ih]
    | add_1 x x' y r ih => dsimp; rw [ih]
    | add_2 x y y' r ih => dsimp; rw [ih]
    | mul_1 x x' y r ih => dsimp; rw [ih]
    | mul_2 x y y' r ih => dsimp; rw [ih]
    | zero_add x => dsimp; rw [zero_add]
    | add_zero x => dsimp; rw [add_zero]
    | one_mul x => dsimp; rw [one_mul]
    | mul_one x => dsimp; rw [mul_one]
    | add_left_neg x => dsimp; rw [add_left_neg]
    | add_comm x y => dsimp; rw [add_comm]
    | mul_comm x y => dsimp; rw [mul_comm]
    

Proof state:
case a.add_assoc
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCat
s : Cocone F
x✝ y✝ x y z : Prequotient F
⊢ descFunLift F s (add (add x y) z) = descFunLift F s (add x (add y z))

Next tactic:
| add_assoc x y z => dsimp; rw [add_assoc]
/-- The function from the colimit commutative ring to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt := by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl => rfl
    | symm x y _ ih => exact ih.symm
    | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2
    | map j j' f x => exact RingHom.congr_fun (s.ι.naturality f) x
    | zero j => simp
    | one j => simp
    | neg j x => simp
    | add j x y => simp
    | mul j x y => simp
    | neg_1 x x' r ih => dsimp; rw [ih]
    | add_1 x x' y r ih => dsimp; rw [ih]
    | add_2 x y y' r ih => dsimp; rw [ih]
    | mul_1 x x' y r ih => dsimp; rw [ih]
    | mul_2 x y y' r ih => dsimp; rw [ih]
    | zero_add x => dsimp; rw [zero_add]
    | add_zero x => dsimp; rw [add_zero]
    | one_mul x => dsimp; rw [one_mul]
    | mul_one x => dsimp; rw [mul_one]
    | add_left_neg x => dsimp; rw [add_left_neg]
    | add_comm x y => dsimp; rw [add_comm]
    | mul_comm x y => dsimp; rw [mul_comm]
    | add_assoc x y z => 

Proof state:
case a.add_assoc
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCat
s : Cocone F
x✝ y✝ x y z : Prequotient F
⊢ descFunLift F s (add (add x y) z) = descFunLift F s (add x (add y z))

Next tactic:
dsimp
/-- The function from the colimit commutative ring to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt := by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl => rfl
    | symm x y _ ih => exact ih.symm
    | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2
    | map j j' f x => exact RingHom.congr_fun (s.ι.naturality f) x
    | zero j => simp
    | one j => simp
    | neg j x => simp
    | add j x y => simp
    | mul j x y => simp
    | neg_1 x x' r ih => dsimp; rw [ih]
    | add_1 x x' y r ih => dsimp; rw [ih]
    | add_2 x y y' r ih => dsimp; rw [ih]
    | mul_1 x x' y r ih => dsimp; rw [ih]
    | mul_2 x y y' r ih => dsimp; rw [ih]
    | zero_add x => dsimp; rw [zero_add]
    | add_zero x => dsimp; rw [add_zero]
    | one_mul x => dsimp; rw [one_mul]
    | mul_one x => dsimp; rw [mul_one]
    | add_left_neg x => dsimp; rw [add_left_neg]
    | add_comm x y => dsimp; rw [add_comm]
    | mul_comm x y => dsimp; rw [mul_comm]
    | add_assoc x y z => dsimp; 

Proof state:
case a.add_assoc
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCat
s : Cocone F
x✝ y✝ x y z : Prequotient F
⊢ descFunLift F s x + descFunLift F s y + descFunLift F s z =
    descFunLift F s x + (descFunLift F s y + descFunLift F s z)

Next tactic:
rw [add_assoc]
/-- The function from the colimit commutative ring to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt := by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl => rfl
    | symm x y _ ih => exact ih.symm
    | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2
    | map j j' f x => exact RingHom.congr_fun (s.ι.naturality f) x
    | zero j => simp
    | one j => simp
    | neg j x => simp
    | add j x y => simp
    | mul j x y => simp
    | neg_1 x x' r ih => dsimp; rw [ih]
    | add_1 x x' y r ih => dsimp; rw [ih]
    | add_2 x y y' r ih => dsimp; rw [ih]
    | mul_1 x x' y r ih => dsimp; rw [ih]
    | mul_2 x y y' r ih => dsimp; rw [ih]
    | zero_add x => dsimp; rw [zero_add]
    | add_zero x => dsimp; rw [add_zero]
    | one_mul x => dsimp; rw [one_mul]
    | mul_one x => dsimp; rw [mul_one]
    | add_left_neg x => dsimp; rw [add_left_neg]
    | add_comm x y => dsimp; rw [add_comm]
    | mul_comm x y => dsimp; rw [mul_comm]
    | add_assoc x y z => dsimp; rw [add_assoc]
    

Proof state:
case a.mul_assoc
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCat
s : Cocone F
x✝ y✝ x y z : Prequotient F
⊢ descFunLift F s (mul (mul x y) z) = descFunLift F s (mul x (mul y z))

Next tactic:
| mul_assoc x y z => dsimp; rw [mul_assoc]
/-- The function from the colimit commutative ring to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt := by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl => rfl
    | symm x y _ ih => exact ih.symm
    | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2
    | map j j' f x => exact RingHom.congr_fun (s.ι.naturality f) x
    | zero j => simp
    | one j => simp
    | neg j x => simp
    | add j x y => simp
    | mul j x y => simp
    | neg_1 x x' r ih => dsimp; rw [ih]
    | add_1 x x' y r ih => dsimp; rw [ih]
    | add_2 x y y' r ih => dsimp; rw [ih]
    | mul_1 x x' y r ih => dsimp; rw [ih]
    | mul_2 x y y' r ih => dsimp; rw [ih]
    | zero_add x => dsimp; rw [zero_add]
    | add_zero x => dsimp; rw [add_zero]
    | one_mul x => dsimp; rw [one_mul]
    | mul_one x => dsimp; rw [mul_one]
    | add_left_neg x => dsimp; rw [add_left_neg]
    | add_comm x y => dsimp; rw [add_comm]
    | mul_comm x y => dsimp; rw [mul_comm]
    | add_assoc x y z => dsimp; rw [add_assoc]
    | mul_assoc x y z => 

Proof state:
case a.mul_assoc
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCat
s : Cocone F
x✝ y✝ x y z : Prequotient F
⊢ descFunLift F s (mul (mul x y) z) = descFunLift F s (mul x (mul y z))

Next tactic:
dsimp
/-- The function from the colimit commutative ring to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt := by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl => rfl
    | symm x y _ ih => exact ih.symm
    | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2
    | map j j' f x => exact RingHom.congr_fun (s.ι.naturality f) x
    | zero j => simp
    | one j => simp
    | neg j x => simp
    | add j x y => simp
    | mul j x y => simp
    | neg_1 x x' r ih => dsimp; rw [ih]
    | add_1 x x' y r ih => dsimp; rw [ih]
    | add_2 x y y' r ih => dsimp; rw [ih]
    | mul_1 x x' y r ih => dsimp; rw [ih]
    | mul_2 x y y' r ih => dsimp; rw [ih]
    | zero_add x => dsimp; rw [zero_add]
    | add_zero x => dsimp; rw [add_zero]
    | one_mul x => dsimp; rw [one_mul]
    | mul_one x => dsimp; rw [mul_one]
    | add_left_neg x => dsimp; rw [add_left_neg]
    | add_comm x y => dsimp; rw [add_comm]
    | mul_comm x y => dsimp; rw [mul_comm]
    | add_assoc x y z => dsimp; rw [add_assoc]
    | mul_assoc x y z => dsimp; 

Proof state:
case a.mul_assoc
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCat
s : Cocone F
x✝ y✝ x y z : Prequotient F
⊢ descFunLift F s x * descFunLift F s y * descFunLift F s z =
    descFunLift F s x * (descFunLift F s y * descFunLift F s z)

Next tactic:
rw [mul_assoc]
/-- The function from the colimit commutative ring to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt := by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl => rfl
    | symm x y _ ih => exact ih.symm
    | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2
    | map j j' f x => exact RingHom.congr_fun (s.ι.naturality f) x
    | zero j => simp
    | one j => simp
    | neg j x => simp
    | add j x y => simp
    | mul j x y => simp
    | neg_1 x x' r ih => dsimp; rw [ih]
    | add_1 x x' y r ih => dsimp; rw [ih]
    | add_2 x y y' r ih => dsimp; rw [ih]
    | mul_1 x x' y r ih => dsimp; rw [ih]
    | mul_2 x y y' r ih => dsimp; rw [ih]
    | zero_add x => dsimp; rw [zero_add]
    | add_zero x => dsimp; rw [add_zero]
    | one_mul x => dsimp; rw [one_mul]
    | mul_one x => dsimp; rw [mul_one]
    | add_left_neg x => dsimp; rw [add_left_neg]
    | add_comm x y => dsimp; rw [add_comm]
    | mul_comm x y => dsimp; rw [mul_comm]
    | add_assoc x y z => dsimp; rw [add_assoc]
    | mul_assoc x y z => dsimp; rw [mul_assoc]
    

Proof state:
case a.left_distrib
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCat
s : Cocone F
x✝ y✝ x y z : Prequotient F
⊢ descFunLift F s (mul x (add y z)) = descFunLift F s (add (mul x y) (mul x z))

Next tactic:
| left_distrib x y z => dsimp; rw [mul_add]
/-- The function from the colimit commutative ring to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt := by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl => rfl
    | symm x y _ ih => exact ih.symm
    | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2
    | map j j' f x => exact RingHom.congr_fun (s.ι.naturality f) x
    | zero j => simp
    | one j => simp
    | neg j x => simp
    | add j x y => simp
    | mul j x y => simp
    | neg_1 x x' r ih => dsimp; rw [ih]
    | add_1 x x' y r ih => dsimp; rw [ih]
    | add_2 x y y' r ih => dsimp; rw [ih]
    | mul_1 x x' y r ih => dsimp; rw [ih]
    | mul_2 x y y' r ih => dsimp; rw [ih]
    | zero_add x => dsimp; rw [zero_add]
    | add_zero x => dsimp; rw [add_zero]
    | one_mul x => dsimp; rw [one_mul]
    | mul_one x => dsimp; rw [mul_one]
    | add_left_neg x => dsimp; rw [add_left_neg]
    | add_comm x y => dsimp; rw [add_comm]
    | mul_comm x y => dsimp; rw [mul_comm]
    | add_assoc x y z => dsimp; rw [add_assoc]
    | mul_assoc x y z => dsimp; rw [mul_assoc]
    | left_distrib x y z => 

Proof state:
case a.left_distrib
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCat
s : Cocone F
x✝ y✝ x y z : Prequotient F
⊢ descFunLift F s (mul x (add y z)) = descFunLift F s (add (mul x y) (mul x z))

Next tactic:
dsimp
/-- The function from the colimit commutative ring to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt := by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl => rfl
    | symm x y _ ih => exact ih.symm
    | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2
    | map j j' f x => exact RingHom.congr_fun (s.ι.naturality f) x
    | zero j => simp
    | one j => simp
    | neg j x => simp
    | add j x y => simp
    | mul j x y => simp
    | neg_1 x x' r ih => dsimp; rw [ih]
    | add_1 x x' y r ih => dsimp; rw [ih]
    | add_2 x y y' r ih => dsimp; rw [ih]
    | mul_1 x x' y r ih => dsimp; rw [ih]
    | mul_2 x y y' r ih => dsimp; rw [ih]
    | zero_add x => dsimp; rw [zero_add]
    | add_zero x => dsimp; rw [add_zero]
    | one_mul x => dsimp; rw [one_mul]
    | mul_one x => dsimp; rw [mul_one]
    | add_left_neg x => dsimp; rw [add_left_neg]
    | add_comm x y => dsimp; rw [add_comm]
    | mul_comm x y => dsimp; rw [mul_comm]
    | add_assoc x y z => dsimp; rw [add_assoc]
    | mul_assoc x y z => dsimp; rw [mul_assoc]
    | left_distrib x y z => dsimp; 

Proof state:
case a.left_distrib
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCat
s : Cocone F
x✝ y✝ x y z : Prequotient F
⊢ descFunLift F s x * (descFunLift F s y + descFunLift F s z) =
    descFunLift F s x * descFunLift F s y + descFunLift F s x * descFunLift F s z

Next tactic:
rw [mul_add]
/-- The function from the colimit commutative ring to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt := by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl => rfl
    | symm x y _ ih => exact ih.symm
    | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2
    | map j j' f x => exact RingHom.congr_fun (s.ι.naturality f) x
    | zero j => simp
    | one j => simp
    | neg j x => simp
    | add j x y => simp
    | mul j x y => simp
    | neg_1 x x' r ih => dsimp; rw [ih]
    | add_1 x x' y r ih => dsimp; rw [ih]
    | add_2 x y y' r ih => dsimp; rw [ih]
    | mul_1 x x' y r ih => dsimp; rw [ih]
    | mul_2 x y y' r ih => dsimp; rw [ih]
    | zero_add x => dsimp; rw [zero_add]
    | add_zero x => dsimp; rw [add_zero]
    | one_mul x => dsimp; rw [one_mul]
    | mul_one x => dsimp; rw [mul_one]
    | add_left_neg x => dsimp; rw [add_left_neg]
    | add_comm x y => dsimp; rw [add_comm]
    | mul_comm x y => dsimp; rw [mul_comm]
    | add_assoc x y z => dsimp; rw [add_assoc]
    | mul_assoc x y z => dsimp; rw [mul_assoc]
    | left_distrib x y z => dsimp; rw [mul_add]
    

Proof state:
case a.right_distrib
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCat
s : Cocone F
x✝ y✝ x y z : Prequotient F
⊢ descFunLift F s (mul (add x y) z) = descFunLift F s (add (mul x z) (mul y z))

Next tactic:
| right_distrib x y z => dsimp; rw [add_mul]
/-- The function from the colimit commutative ring to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt := by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl => rfl
    | symm x y _ ih => exact ih.symm
    | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2
    | map j j' f x => exact RingHom.congr_fun (s.ι.naturality f) x
    | zero j => simp
    | one j => simp
    | neg j x => simp
    | add j x y => simp
    | mul j x y => simp
    | neg_1 x x' r ih => dsimp; rw [ih]
    | add_1 x x' y r ih => dsimp; rw [ih]
    | add_2 x y y' r ih => dsimp; rw [ih]
    | mul_1 x x' y r ih => dsimp; rw [ih]
    | mul_2 x y y' r ih => dsimp; rw [ih]
    | zero_add x => dsimp; rw [zero_add]
    | add_zero x => dsimp; rw [add_zero]
    | one_mul x => dsimp; rw [one_mul]
    | mul_one x => dsimp; rw [mul_one]
    | add_left_neg x => dsimp; rw [add_left_neg]
    | add_comm x y => dsimp; rw [add_comm]
    | mul_comm x y => dsimp; rw [mul_comm]
    | add_assoc x y z => dsimp; rw [add_assoc]
    | mul_assoc x y z => dsimp; rw [mul_assoc]
    | left_distrib x y z => dsimp; rw [mul_add]
    | right_distrib x y z => 

Proof state:
case a.right_distrib
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCat
s : Cocone F
x✝ y✝ x y z : Prequotient F
⊢ descFunLift F s (mul (add x y) z) = descFunLift F s (add (mul x z) (mul y z))

Next tactic:
dsimp
/-- The function from the colimit commutative ring to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt := by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl => rfl
    | symm x y _ ih => exact ih.symm
    | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2
    | map j j' f x => exact RingHom.congr_fun (s.ι.naturality f) x
    | zero j => simp
    | one j => simp
    | neg j x => simp
    | add j x y => simp
    | mul j x y => simp
    | neg_1 x x' r ih => dsimp; rw [ih]
    | add_1 x x' y r ih => dsimp; rw [ih]
    | add_2 x y y' r ih => dsimp; rw [ih]
    | mul_1 x x' y r ih => dsimp; rw [ih]
    | mul_2 x y y' r ih => dsimp; rw [ih]
    | zero_add x => dsimp; rw [zero_add]
    | add_zero x => dsimp; rw [add_zero]
    | one_mul x => dsimp; rw [one_mul]
    | mul_one x => dsimp; rw [mul_one]
    | add_left_neg x => dsimp; rw [add_left_neg]
    | add_comm x y => dsimp; rw [add_comm]
    | mul_comm x y => dsimp; rw [mul_comm]
    | add_assoc x y z => dsimp; rw [add_assoc]
    | mul_assoc x y z => dsimp; rw [mul_assoc]
    | left_distrib x y z => dsimp; rw [mul_add]
    | right_distrib x y z => dsimp; 

Proof state:
case a.right_distrib
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCat
s : Cocone F
x✝ y✝ x y z : Prequotient F
⊢ (descFunLift F s x + descFunLift F s y) * descFunLift F s z =
    descFunLift F s x * descFunLift F s z + descFunLift F s y * descFunLift F s z

Next tactic:
rw [add_mul]
/-- The function from the colimit commutative ring to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt := by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl => rfl
    | symm x y _ ih => exact ih.symm
    | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2
    | map j j' f x => exact RingHom.congr_fun (s.ι.naturality f) x
    | zero j => simp
    | one j => simp
    | neg j x => simp
    | add j x y => simp
    | mul j x y => simp
    | neg_1 x x' r ih => dsimp; rw [ih]
    | add_1 x x' y r ih => dsimp; rw [ih]
    | add_2 x y y' r ih => dsimp; rw [ih]
    | mul_1 x x' y r ih => dsimp; rw [ih]
    | mul_2 x y y' r ih => dsimp; rw [ih]
    | zero_add x => dsimp; rw [zero_add]
    | add_zero x => dsimp; rw [add_zero]
    | one_mul x => dsimp; rw [one_mul]
    | mul_one x => dsimp; rw [mul_one]
    | add_left_neg x => dsimp; rw [add_left_neg]
    | add_comm x y => dsimp; rw [add_comm]
    | mul_comm x y => dsimp; rw [mul_comm]
    | add_assoc x y z => dsimp; rw [add_assoc]
    | mul_assoc x y z => dsimp; rw [mul_assoc]
    | left_distrib x y z => dsimp; rw [mul_add]
    | right_distrib x y z => dsimp; rw [add_mul]
    

Proof state:
case a.zero_mul
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCat
s : Cocone F
x✝ y x : Prequotient F
⊢ descFunLift F s (mul zero x) = descFunLift F s zero

Next tactic:
| zero_mul x => dsimp; rw [zero_mul]
/-- The function from the colimit commutative ring to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt := by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl => rfl
    | symm x y _ ih => exact ih.symm
    | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2
    | map j j' f x => exact RingHom.congr_fun (s.ι.naturality f) x
    | zero j => simp
    | one j => simp
    | neg j x => simp
    | add j x y => simp
    | mul j x y => simp
    | neg_1 x x' r ih => dsimp; rw [ih]
    | add_1 x x' y r ih => dsimp; rw [ih]
    | add_2 x y y' r ih => dsimp; rw [ih]
    | mul_1 x x' y r ih => dsimp; rw [ih]
    | mul_2 x y y' r ih => dsimp; rw [ih]
    | zero_add x => dsimp; rw [zero_add]
    | add_zero x => dsimp; rw [add_zero]
    | one_mul x => dsimp; rw [one_mul]
    | mul_one x => dsimp; rw [mul_one]
    | add_left_neg x => dsimp; rw [add_left_neg]
    | add_comm x y => dsimp; rw [add_comm]
    | mul_comm x y => dsimp; rw [mul_comm]
    | add_assoc x y z => dsimp; rw [add_assoc]
    | mul_assoc x y z => dsimp; rw [mul_assoc]
    | left_distrib x y z => dsimp; rw [mul_add]
    | right_distrib x y z => dsimp; rw [add_mul]
    | zero_mul x => 

Proof state:
case a.zero_mul
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCat
s : Cocone F
x✝ y x : Prequotient F
⊢ descFunLift F s (mul zero x) = descFunLift F s zero

Next tactic:
dsimp
/-- The function from the colimit commutative ring to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt := by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl => rfl
    | symm x y _ ih => exact ih.symm
    | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2
    | map j j' f x => exact RingHom.congr_fun (s.ι.naturality f) x
    | zero j => simp
    | one j => simp
    | neg j x => simp
    | add j x y => simp
    | mul j x y => simp
    | neg_1 x x' r ih => dsimp; rw [ih]
    | add_1 x x' y r ih => dsimp; rw [ih]
    | add_2 x y y' r ih => dsimp; rw [ih]
    | mul_1 x x' y r ih => dsimp; rw [ih]
    | mul_2 x y y' r ih => dsimp; rw [ih]
    | zero_add x => dsimp; rw [zero_add]
    | add_zero x => dsimp; rw [add_zero]
    | one_mul x => dsimp; rw [one_mul]
    | mul_one x => dsimp; rw [mul_one]
    | add_left_neg x => dsimp; rw [add_left_neg]
    | add_comm x y => dsimp; rw [add_comm]
    | mul_comm x y => dsimp; rw [mul_comm]
    | add_assoc x y z => dsimp; rw [add_assoc]
    | mul_assoc x y z => dsimp; rw [mul_assoc]
    | left_distrib x y z => dsimp; rw [mul_add]
    | right_distrib x y z => dsimp; rw [add_mul]
    | zero_mul x => dsimp; 

Proof state:
case a.zero_mul
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCat
s : Cocone F
x✝ y x : Prequotient F
⊢ 0 * descFunLift F s x = 0

Next tactic:
rw [zero_mul]
/-- The function from the colimit commutative ring to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt := by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl => rfl
    | symm x y _ ih => exact ih.symm
    | trans x y z _ _ ih1 ih2 => exact ih1.trans ih2
    | map j j' f x => exact RingHom.congr_fun (s.ι.naturality f) x
    | zero j => simp
    | one j => simp
    | neg j x => simp
    | add j x y => simp
    | mul j x y => simp
    | neg_1 x x' r ih => dsimp; rw [ih]
    | add_1 x x' y r ih => dsimp; rw [ih]
    | add_2 x y y' r ih => dsimp; rw [ih]
    | mul_1 x x' y r ih => dsimp; rw [ih]
    | mul_2 x y y' r ih => dsimp; rw [ih]
    | zero_add x => dsimp; rw [zero_add]
    | add_zero x => dsimp; rw [add_zero]
    | one_mul x => dsimp; rw [one_mul]
    | mul_one x => dsimp; rw [mul_one]
    | add_left_neg x => dsimp; rw [add_left_neg]
    | add_comm x y => dsimp; rw [add_comm]
    | mul_comm x y => dsimp; rw [mul_comm]
    | add_assoc x y z => dsimp; rw [add_assoc]
    | mul_assoc x y z => dsimp; rw [mul_assoc]
    | left_distrib x y z => dsimp; rw [mul_add]
    | right_distrib x y z => dsimp; rw [add_mul]
    | zero_mul x => dsimp; rw [zero_mul]
    

Proof state:
case a.mul_zero
J : Type v
inst✝ : SmallCategory J
F : J ⥤ CommRingCat
s : Cocone F
x✝ y x : Prequotient F
⊢ descFunLift F s (mul x zero) = descFunLift F s zero

Next tactic:
| mul_zero x => dsimp; rw [mul_zero]