l3lab/ntp-mathlib huggingface dataset:
instance (priority := 100) wfDvdMonoid {R : Type*} [CommRing R] [IsDomain R] [WfDvdMonoid R] :
    WfDvdMonoid R[X] where
  wellFounded_dvdNotUnit := by
    classical
      refine'
        RelHomClass.wellFounded
          (⟨fun p : R[X] =>
              ((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), _⟩ :
            DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wellFounded_dvdNotUnit)
      rintro a b ⟨ane0, ⟨c, ⟨not_unit_c, rfl⟩⟩⟩
      dsimp
      rw [Polynomial.degree_mul, if_neg ane0]
      split_ifs with hac
      · 

Proof state:
case pos
R✝ : Type u
S : Type u_1
σ : Type v
M : Type w
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R✝ M
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
a : R[X]
ane0 : a ≠ 0
c : R[X]
not_unit_c : ¬IsUnit c
hac : a * c = 0
⊢ Prod.Lex (fun x x_1 => x < x_1) DvdNotUnit (↑(degree a), leadingCoeff a) (⊤, leadingCoeff (a * c))

Next tactic:
rw [hac, Polynomial.leadingCoeff_zero]
instance (priority := 100) wfDvdMonoid {R : Type*} [CommRing R] [IsDomain R] [WfDvdMonoid R] :
    WfDvdMonoid R[X] where
  wellFounded_dvdNotUnit := by
    classical
      refine'
        RelHomClass.wellFounded
          (⟨fun p : R[X] =>
              ((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), _⟩ :
            DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wellFounded_dvdNotUnit)
      rintro a b ⟨ane0, ⟨c, ⟨not_unit_c, rfl⟩⟩⟩
      dsimp
      rw [Polynomial.degree_mul, if_neg ane0]
      split_ifs with hac
      · rw [hac, Polynomial.leadingCoeff_zero]
        

Proof state:
case pos
R✝ : Type u
S : Type u_1
σ : Type v
M : Type w
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R✝ M
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
a : R[X]
ane0 : a ≠ 0
c : R[X]
not_unit_c : ¬IsUnit c
hac : a * c = 0
⊢ Prod.Lex (fun x x_1 => x < x_1) DvdNotUnit (↑(degree a), leadingCoeff a) (⊤, 0)

Next tactic:
apply Prod.Lex.left
instance (priority := 100) wfDvdMonoid {R : Type*} [CommRing R] [IsDomain R] [WfDvdMonoid R] :
    WfDvdMonoid R[X] where
  wellFounded_dvdNotUnit := by
    classical
      refine'
        RelHomClass.wellFounded
          (⟨fun p : R[X] =>
              ((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), _⟩ :
            DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wellFounded_dvdNotUnit)
      rintro a b ⟨ane0, ⟨c, ⟨not_unit_c, rfl⟩⟩⟩
      dsimp
      rw [Polynomial.degree_mul, if_neg ane0]
      split_ifs with hac
      · rw [hac, Polynomial.leadingCoeff_zero]
        apply Prod.Lex.left
        

Proof state:
case pos.h
R✝ : Type u
S : Type u_1
σ : Type v
M : Type w
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R✝ M
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
a : R[X]
ane0 : a ≠ 0
c : R[X]
not_unit_c : ¬IsUnit c
hac : a * c = 0
⊢ ↑(degree a) < ⊤

Next tactic:
exact lt_of_le_of_ne le_top WithTop.coe_ne_top
instance (priority := 100) wfDvdMonoid {R : Type*} [CommRing R] [IsDomain R] [WfDvdMonoid R] :
    WfDvdMonoid R[X] where
  wellFounded_dvdNotUnit := by
    classical
      refine'
        RelHomClass.wellFounded
          (⟨fun p : R[X] =>
              ((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), _⟩ :
            DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wellFounded_dvdNotUnit)
      rintro a b ⟨ane0, ⟨c, ⟨not_unit_c, rfl⟩⟩⟩
      dsimp
      rw [Polynomial.degree_mul, if_neg ane0]
      split_ifs with hac
      · rw [hac, Polynomial.leadingCoeff_zero]
        apply Prod.Lex.left
        exact lt_of_le_of_ne le_top WithTop.coe_ne_top
      

Proof state:
case neg
R✝ : Type u
S : Type u_1
σ : Type v
M : Type w
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R✝ M
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
a : R[X]
ane0 : a ≠ 0
c : R[X]
not_unit_c : ¬IsUnit c
hac : ¬a * c = 0
⊢ Prod.Lex (fun x x_1 => x < x_1) DvdNotUnit (↑(degree a), leadingCoeff a)
    (↑(degree a + degree c), leadingCoeff (a * c))

Next tactic:
have cne0 : c ≠ 0 := right_ne_zero_of_mul hac
instance (priority := 100) wfDvdMonoid {R : Type*} [CommRing R] [IsDomain R] [WfDvdMonoid R] :
    WfDvdMonoid R[X] where
  wellFounded_dvdNotUnit := by
    classical
      refine'
        RelHomClass.wellFounded
          (⟨fun p : R[X] =>
              ((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), _⟩ :
            DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wellFounded_dvdNotUnit)
      rintro a b ⟨ane0, ⟨c, ⟨not_unit_c, rfl⟩⟩⟩
      dsimp
      rw [Polynomial.degree_mul, if_neg ane0]
      split_ifs with hac
      · rw [hac, Polynomial.leadingCoeff_zero]
        apply Prod.Lex.left
        exact lt_of_le_of_ne le_top WithTop.coe_ne_top
      have cne0 : c ≠ 0 := right_ne_zero_of_mul hac
      

Proof state:
case neg
R✝ : Type u
S : Type u_1
σ : Type v
M : Type w
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R✝ M
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
a : R[X]
ane0 : a ≠ 0
c : R[X]
not_unit_c : ¬IsUnit c
hac : ¬a * c = 0
cne0 : c ≠ 0
⊢ Prod.Lex (fun x x_1 => x < x_1) DvdNotUnit (↑(degree a), leadingCoeff a)
    (↑(degree a + degree c), leadingCoeff (a * c))

Next tactic:
simp only [cne0, ane0, Polynomial.leadingCoeff_mul]
instance (priority := 100) wfDvdMonoid {R : Type*} [CommRing R] [IsDomain R] [WfDvdMonoid R] :
    WfDvdMonoid R[X] where
  wellFounded_dvdNotUnit := by
    classical
      refine'
        RelHomClass.wellFounded
          (⟨fun p : R[X] =>
              ((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), _⟩ :
            DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wellFounded_dvdNotUnit)
      rintro a b ⟨ane0, ⟨c, ⟨not_unit_c, rfl⟩⟩⟩
      dsimp
      rw [Polynomial.degree_mul, if_neg ane0]
      split_ifs with hac
      · rw [hac, Polynomial.leadingCoeff_zero]
        apply Prod.Lex.left
        exact lt_of_le_of_ne le_top WithTop.coe_ne_top
      have cne0 : c ≠ 0 := right_ne_zero_of_mul hac
      simp only [cne0, ane0, Polynomial.leadingCoeff_mul]
      

Proof state:
case neg
R✝ : Type u
S : Type u_1
σ : Type v
M : Type w
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R✝ M
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
a : R[X]
ane0 : a ≠ 0
c : R[X]
not_unit_c : ¬IsUnit c
hac : ¬a * c = 0
cne0 : c ≠ 0
⊢ Prod.Lex (fun x x_1 => x < x_1) DvdNotUnit (↑(degree a), leadingCoeff a)
    (↑(degree a + degree c), leadingCoeff a * leadingCoeff c)

Next tactic:
by_cases hdeg : c.degree = 0
instance (priority := 100) wfDvdMonoid {R : Type*} [CommRing R] [IsDomain R] [WfDvdMonoid R] :
    WfDvdMonoid R[X] where
  wellFounded_dvdNotUnit := by
    classical
      refine'
        RelHomClass.wellFounded
          (⟨fun p : R[X] =>
              ((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), _⟩ :
            DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wellFounded_dvdNotUnit)
      rintro a b ⟨ane0, ⟨c, ⟨not_unit_c, rfl⟩⟩⟩
      dsimp
      rw [Polynomial.degree_mul, if_neg ane0]
      split_ifs with hac
      · rw [hac, Polynomial.leadingCoeff_zero]
        apply Prod.Lex.left
        exact lt_of_le_of_ne le_top WithTop.coe_ne_top
      have cne0 : c ≠ 0 := right_ne_zero_of_mul hac
      simp only [cne0, ane0, Polynomial.leadingCoeff_mul]
      by_cases hdeg : c.degree = 0
      · 

Proof state:
case pos
R✝ : Type u
S : Type u_1
σ : Type v
M : Type w
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R✝ M
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
a : R[X]
ane0 : a ≠ 0
c : R[X]
not_unit_c : ¬IsUnit c
hac : ¬a * c = 0
cne0 : c ≠ 0
hdeg : degree c = 0
⊢ Prod.Lex (fun x x_1 => x < x_1) DvdNotUnit (↑(degree a), leadingCoeff a)
    (↑(degree a + degree c), leadingCoeff a * leadingCoeff c)

Next tactic:
simp only [hdeg, add_zero]
instance (priority := 100) wfDvdMonoid {R : Type*} [CommRing R] [IsDomain R] [WfDvdMonoid R] :
    WfDvdMonoid R[X] where
  wellFounded_dvdNotUnit := by
    classical
      refine'
        RelHomClass.wellFounded
          (⟨fun p : R[X] =>
              ((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), _⟩ :
            DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wellFounded_dvdNotUnit)
      rintro a b ⟨ane0, ⟨c, ⟨not_unit_c, rfl⟩⟩⟩
      dsimp
      rw [Polynomial.degree_mul, if_neg ane0]
      split_ifs with hac
      · rw [hac, Polynomial.leadingCoeff_zero]
        apply Prod.Lex.left
        exact lt_of_le_of_ne le_top WithTop.coe_ne_top
      have cne0 : c ≠ 0 := right_ne_zero_of_mul hac
      simp only [cne0, ane0, Polynomial.leadingCoeff_mul]
      by_cases hdeg : c.degree = 0
      · simp only [hdeg, add_zero]
        

Proof state:
case pos
R✝ : Type u
S : Type u_1
σ : Type v
M : Type w
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R✝ M
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
a : R[X]
ane0 : a ≠ 0
c : R[X]
not_unit_c : ¬IsUnit c
hac : ¬a * c = 0
cne0 : c ≠ 0
hdeg : degree c = 0
⊢ Prod.Lex (fun x x_1 => x < x_1) DvdNotUnit (↑(degree a), leadingCoeff a)
    (↑(degree a), leadingCoeff a * leadingCoeff c)

Next tactic:
refine' Prod.Lex.right _ ⟨_, ⟨c.leadingCoeff, fun unit_c => not_unit_c _, rfl⟩⟩
instance (priority := 100) wfDvdMonoid {R : Type*} [CommRing R] [IsDomain R] [WfDvdMonoid R] :
    WfDvdMonoid R[X] where
  wellFounded_dvdNotUnit := by
    classical
      refine'
        RelHomClass.wellFounded
          (⟨fun p : R[X] =>
              ((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), _⟩ :
            DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wellFounded_dvdNotUnit)
      rintro a b ⟨ane0, ⟨c, ⟨not_unit_c, rfl⟩⟩⟩
      dsimp
      rw [Polynomial.degree_mul, if_neg ane0]
      split_ifs with hac
      · rw [hac, Polynomial.leadingCoeff_zero]
        apply Prod.Lex.left
        exact lt_of_le_of_ne le_top WithTop.coe_ne_top
      have cne0 : c ≠ 0 := right_ne_zero_of_mul hac
      simp only [cne0, ane0, Polynomial.leadingCoeff_mul]
      by_cases hdeg : c.degree = 0
      · simp only [hdeg, add_zero]
        refine' Prod.Lex.right _ ⟨_, ⟨c.leadingCoeff, fun unit_c => not_unit_c _, rfl⟩⟩
        · 

Proof state:
case pos.refine'_1
R✝ : Type u
S : Type u_1
σ : Type v
M : Type w
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R✝ M
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
a : R[X]
ane0 : a ≠ 0
c : R[X]
not_unit_c : ¬IsUnit c
hac : ¬a * c = 0
cne0 : c ≠ 0
hdeg : degree c = 0
⊢ leadingCoeff a ≠ 0

Next tactic:
rwa [Ne, Polynomial.leadingCoeff_eq_zero]
instance (priority := 100) wfDvdMonoid {R : Type*} [CommRing R] [IsDomain R] [WfDvdMonoid R] :
    WfDvdMonoid R[X] where
  wellFounded_dvdNotUnit := by
    classical
      refine'
        RelHomClass.wellFounded
          (⟨fun p : R[X] =>
              ((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), _⟩ :
            DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wellFounded_dvdNotUnit)
      rintro a b ⟨ane0, ⟨c, ⟨not_unit_c, rfl⟩⟩⟩
      dsimp
      rw [Polynomial.degree_mul, if_neg ane0]
      split_ifs with hac
      · rw [hac, Polynomial.leadingCoeff_zero]
        apply Prod.Lex.left
        exact lt_of_le_of_ne le_top WithTop.coe_ne_top
      have cne0 : c ≠ 0 := right_ne_zero_of_mul hac
      simp only [cne0, ane0, Polynomial.leadingCoeff_mul]
      by_cases hdeg : c.degree = 0
      · simp only [hdeg, add_zero]
        refine' Prod.Lex.right _ ⟨_, ⟨c.leadingCoeff, fun unit_c => not_unit_c _, rfl⟩⟩
        · rwa [Ne, Polynomial.leadingCoeff_eq_zero]
        

Proof state:
case pos.refine'_2
R✝ : Type u
S : Type u_1
σ : Type v
M : Type w
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R✝ M
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
a : R[X]
ane0 : a ≠ 0
c : R[X]
not_unit_c : ¬IsUnit c
hac : ¬a * c = 0
cne0 : c ≠ 0
hdeg : degree c = 0
unit_c : IsUnit (leadingCoeff c)
⊢ IsUnit c

Next tactic:
rw [Polynomial.isUnit_iff, Polynomial.eq_C_of_degree_eq_zero hdeg]
instance (priority := 100) wfDvdMonoid {R : Type*} [CommRing R] [IsDomain R] [WfDvdMonoid R] :
    WfDvdMonoid R[X] where
  wellFounded_dvdNotUnit := by
    classical
      refine'
        RelHomClass.wellFounded
          (⟨fun p : R[X] =>
              ((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), _⟩ :
            DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wellFounded_dvdNotUnit)
      rintro a b ⟨ane0, ⟨c, ⟨not_unit_c, rfl⟩⟩⟩
      dsimp
      rw [Polynomial.degree_mul, if_neg ane0]
      split_ifs with hac
      · rw [hac, Polynomial.leadingCoeff_zero]
        apply Prod.Lex.left
        exact lt_of_le_of_ne le_top WithTop.coe_ne_top
      have cne0 : c ≠ 0 := right_ne_zero_of_mul hac
      simp only [cne0, ane0, Polynomial.leadingCoeff_mul]
      by_cases hdeg : c.degree = 0
      · simp only [hdeg, add_zero]
        refine' Prod.Lex.right _ ⟨_, ⟨c.leadingCoeff, fun unit_c => not_unit_c _, rfl⟩⟩
        · rwa [Ne, Polynomial.leadingCoeff_eq_zero]
        rw [Polynomial.isUnit_iff, Polynomial.eq_C_of_degree_eq_zero hdeg]
        

Proof state:
case pos.refine'_2
R✝ : Type u
S : Type u_1
σ : Type v
M : Type w
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R✝ M
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
a : R[X]
ane0 : a ≠ 0
c : R[X]
not_unit_c : ¬IsUnit c
hac : ¬a * c = 0
cne0 : c ≠ 0
hdeg : degree c = 0
unit_c : IsUnit (leadingCoeff c)
⊢ ∃ r, IsUnit r ∧ C r = C (coeff c 0)

Next tactic:
use c.leadingCoeff, unit_c
instance (priority := 100) wfDvdMonoid {R : Type*} [CommRing R] [IsDomain R] [WfDvdMonoid R] :
    WfDvdMonoid R[X] where
  wellFounded_dvdNotUnit := by
    classical
      refine'
        RelHomClass.wellFounded
          (⟨fun p : R[X] =>
              ((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), _⟩ :
            DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wellFounded_dvdNotUnit)
      rintro a b ⟨ane0, ⟨c, ⟨not_unit_c, rfl⟩⟩⟩
      dsimp
      rw [Polynomial.degree_mul, if_neg ane0]
      split_ifs with hac
      · rw [hac, Polynomial.leadingCoeff_zero]
        apply Prod.Lex.left
        exact lt_of_le_of_ne le_top WithTop.coe_ne_top
      have cne0 : c ≠ 0 := right_ne_zero_of_mul hac
      simp only [cne0, ane0, Polynomial.leadingCoeff_mul]
      by_cases hdeg : c.degree = 0
      · simp only [hdeg, add_zero]
        refine' Prod.Lex.right _ ⟨_, ⟨c.leadingCoeff, fun unit_c => not_unit_c _, rfl⟩⟩
        · rwa [Ne, Polynomial.leadingCoeff_eq_zero]
        rw [Polynomial.isUnit_iff, Polynomial.eq_C_of_degree_eq_zero hdeg]
        use c.leadingCoeff, unit_c
        

Proof state:
case right
R✝ : Type u
S : Type u_1
σ : Type v
M : Type w
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R✝ M
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
a : R[X]
ane0 : a ≠ 0
c : R[X]
not_unit_c : ¬IsUnit c
hac : ¬a * c = 0
cne0 : c ≠ 0
hdeg : degree c = 0
unit_c : IsUnit (leadingCoeff c)
⊢ C (leadingCoeff c) = C (coeff c 0)

Next tactic:
rw [Polynomial.leadingCoeff, Polynomial.natDegree_eq_of_degree_eq_some hdeg]
instance (priority := 100) wfDvdMonoid {R : Type*} [CommRing R] [IsDomain R] [WfDvdMonoid R] :
    WfDvdMonoid R[X] where
  wellFounded_dvdNotUnit := by
    classical
      refine'
        RelHomClass.wellFounded
          (⟨fun p : R[X] =>
              ((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), _⟩ :
            DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wellFounded_dvdNotUnit)
      rintro a b ⟨ane0, ⟨c, ⟨not_unit_c, rfl⟩⟩⟩
      dsimp
      rw [Polynomial.degree_mul, if_neg ane0]
      split_ifs with hac
      · rw [hac, Polynomial.leadingCoeff_zero]
        apply Prod.Lex.left
        exact lt_of_le_of_ne le_top WithTop.coe_ne_top
      have cne0 : c ≠ 0 := right_ne_zero_of_mul hac
      simp only [cne0, ane0, Polynomial.leadingCoeff_mul]
      by_cases hdeg : c.degree = 0
      · simp only [hdeg, add_zero]
        refine' Prod.Lex.right _ ⟨_, ⟨c.leadingCoeff, fun unit_c => not_unit_c _, rfl⟩⟩
        · rwa [Ne, Polynomial.leadingCoeff_eq_zero]
        rw [Polynomial.isUnit_iff, Polynomial.eq_C_of_degree_eq_zero hdeg]
        use c.leadingCoeff, unit_c
        rw [Polynomial.leadingCoeff, Polynomial.natDegree_eq_of_degree_eq_some hdeg]; 

Proof state:
case right
R✝ : Type u
S : Type u_1
σ : Type v
M : Type w
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R✝ M
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
a : R[X]
ane0 : a ≠ 0
c : R[X]
not_unit_c : ¬IsUnit c
hac : ¬a * c = 0
cne0 : c ≠ 0
hdeg : degree c = 0
unit_c : IsUnit (leadingCoeff c)
⊢ C (coeff c Zero.zero) = C (coeff c 0)

Next tactic:
rfl
instance (priority := 100) wfDvdMonoid {R : Type*} [CommRing R] [IsDomain R] [WfDvdMonoid R] :
    WfDvdMonoid R[X] where
  wellFounded_dvdNotUnit := by
    classical
      refine'
        RelHomClass.wellFounded
          (⟨fun p : R[X] =>
              ((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), _⟩ :
            DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wellFounded_dvdNotUnit)
      rintro a b ⟨ane0, ⟨c, ⟨not_unit_c, rfl⟩⟩⟩
      dsimp
      rw [Polynomial.degree_mul, if_neg ane0]
      split_ifs with hac
      · rw [hac, Polynomial.leadingCoeff_zero]
        apply Prod.Lex.left
        exact lt_of_le_of_ne le_top WithTop.coe_ne_top
      have cne0 : c ≠ 0 := right_ne_zero_of_mul hac
      simp only [cne0, ane0, Polynomial.leadingCoeff_mul]
      by_cases hdeg : c.degree = 0
      · simp only [hdeg, add_zero]
        refine' Prod.Lex.right _ ⟨_, ⟨c.leadingCoeff, fun unit_c => not_unit_c _, rfl⟩⟩
        · rwa [Ne, Polynomial.leadingCoeff_eq_zero]
        rw [Polynomial.isUnit_iff, Polynomial.eq_C_of_degree_eq_zero hdeg]
        use c.leadingCoeff, unit_c
        rw [Polynomial.leadingCoeff, Polynomial.natDegree_eq_of_degree_eq_some hdeg]; rfl
      · 

Proof state:
case neg
R✝ : Type u
S : Type u_1
σ : Type v
M : Type w
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R✝ M
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
a : R[X]
ane0 : a ≠ 0
c : R[X]
not_unit_c : ¬IsUnit c
hac : ¬a * c = 0
cne0 : c ≠ 0
hdeg : ¬degree c = 0
⊢ Prod.Lex (fun x x_1 => x < x_1) DvdNotUnit (↑(degree a), leadingCoeff a)
    (↑(degree a + degree c), leadingCoeff a * leadingCoeff c)

Next tactic:
apply Prod.Lex.left
instance (priority := 100) wfDvdMonoid {R : Type*} [CommRing R] [IsDomain R] [WfDvdMonoid R] :
    WfDvdMonoid R[X] where
  wellFounded_dvdNotUnit := by
    classical
      refine'
        RelHomClass.wellFounded
          (⟨fun p : R[X] =>
              ((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), _⟩ :
            DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wellFounded_dvdNotUnit)
      rintro a b ⟨ane0, ⟨c, ⟨not_unit_c, rfl⟩⟩⟩
      dsimp
      rw [Polynomial.degree_mul, if_neg ane0]
      split_ifs with hac
      · rw [hac, Polynomial.leadingCoeff_zero]
        apply Prod.Lex.left
        exact lt_of_le_of_ne le_top WithTop.coe_ne_top
      have cne0 : c ≠ 0 := right_ne_zero_of_mul hac
      simp only [cne0, ane0, Polynomial.leadingCoeff_mul]
      by_cases hdeg : c.degree = 0
      · simp only [hdeg, add_zero]
        refine' Prod.Lex.right _ ⟨_, ⟨c.leadingCoeff, fun unit_c => not_unit_c _, rfl⟩⟩
        · rwa [Ne, Polynomial.leadingCoeff_eq_zero]
        rw [Polynomial.isUnit_iff, Polynomial.eq_C_of_degree_eq_zero hdeg]
        use c.leadingCoeff, unit_c
        rw [Polynomial.leadingCoeff, Polynomial.natDegree_eq_of_degree_eq_some hdeg]; rfl
      · apply Prod.Lex.left
        

Proof state:
case neg.h
R✝ : Type u
S : Type u_1
σ : Type v
M : Type w
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R✝ M
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
a : R[X]
ane0 : a ≠ 0
c : R[X]
not_unit_c : ¬IsUnit c
hac : ¬a * c = 0
cne0 : c ≠ 0
hdeg : ¬degree c = 0
⊢ ↑(degree a) < ↑(degree a + degree c)

Next tactic:
rw [Polynomial.degree_eq_natDegree cne0] at *
instance (priority := 100) wfDvdMonoid {R : Type*} [CommRing R] [IsDomain R] [WfDvdMonoid R] :
    WfDvdMonoid R[X] where
  wellFounded_dvdNotUnit := by
    classical
      refine'
        RelHomClass.wellFounded
          (⟨fun p : R[X] =>
              ((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), _⟩ :
            DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wellFounded_dvdNotUnit)
      rintro a b ⟨ane0, ⟨c, ⟨not_unit_c, rfl⟩⟩⟩
      dsimp
      rw [Polynomial.degree_mul, if_neg ane0]
      split_ifs with hac
      · rw [hac, Polynomial.leadingCoeff_zero]
        apply Prod.Lex.left
        exact lt_of_le_of_ne le_top WithTop.coe_ne_top
      have cne0 : c ≠ 0 := right_ne_zero_of_mul hac
      simp only [cne0, ane0, Polynomial.leadingCoeff_mul]
      by_cases hdeg : c.degree = 0
      · simp only [hdeg, add_zero]
        refine' Prod.Lex.right _ ⟨_, ⟨c.leadingCoeff, fun unit_c => not_unit_c _, rfl⟩⟩
        · rwa [Ne, Polynomial.leadingCoeff_eq_zero]
        rw [Polynomial.isUnit_iff, Polynomial.eq_C_of_degree_eq_zero hdeg]
        use c.leadingCoeff, unit_c
        rw [Polynomial.leadingCoeff, Polynomial.natDegree_eq_of_degree_eq_some hdeg]; rfl
      · apply Prod.Lex.left
        rw [Polynomial.degree_eq_natDegree cne0] at *
        

Proof state:
case neg.h
R✝ : Type u
S : Type u_1
σ : Type v
M : Type w
inst✝⁶ : CommRing R✝
inst✝⁵ : CommRing S
inst✝⁴ : AddCommGroup M
inst✝³ : Module R✝ M
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : WfDvdMonoid R
a : R[X]
ane0 : a ≠ 0
c : R[X]
not_unit_c : ¬IsUnit c
hac : ¬a * c = 0
cne0 : c ≠ 0
hdeg : ¬↑(natDegree c) = 0
⊢ ↑(degree a) < ↑(degree a + ↑(natDegree c))

Next tactic:
rw [WithTop.coe_lt_coe, Polynomial.degree_eq_natDegree ane0, ← Nat.cast_add, Nat.cast_lt]