l3lab/ntp-mathlib huggingface dataset:
theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : multiplicity.Finite p q :=
  have zn0 : (0 : R) ≠ 1 :=
    haveI := Nontrivial.of_polynomial_ne hq
    zero_ne_one
  ⟨natDegree q, fun ⟨r, hr⟩ => by
    have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at hp
    have hr0 : r ≠ 0 := fun hr0 => by subst hr0; simp [hq] at hr
    have hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 := by simp [show _ = _ from hmp]
    have hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0 := hpn1.symm ▸ zn0.symm
    

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
r : R[X]
hr : q = p ^ (natDegree q + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1
hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0
⊢ False

Next tactic:
have hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0 := by
      simp only [leadingCoeff_pow' hpn0', leadingCoeff_eq_zero, hpn1, one_pow, one_mul, Ne.def,
          hr0, not_false_eq_true]
theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : multiplicity.Finite p q :=
  have zn0 : (0 : R) ≠ 1 :=
    haveI := Nontrivial.of_polynomial_ne hq
    zero_ne_one
  ⟨natDegree q, fun ⟨r, hr⟩ => by
    have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at hp
    have hr0 : r ≠ 0 := fun hr0 => by subst hr0; simp [hq] at hr
    have hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 := by simp [show _ = _ from hmp]
    have hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0 := hpn1.symm ▸ zn0.symm
    have hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0 := by
      

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
r : R[X]
hr : q = p ^ (natDegree q + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1
hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0
⊢ leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0

Next tactic:
simp only [leadingCoeff_pow' hpn0', leadingCoeff_eq_zero, hpn1, one_pow, one_mul, Ne.def,
          hr0, not_false_eq_true]
theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : multiplicity.Finite p q :=
  have zn0 : (0 : R) ≠ 1 :=
    haveI := Nontrivial.of_polynomial_ne hq
    zero_ne_one
  ⟨natDegree q, fun ⟨r, hr⟩ => by
    have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at hp
    have hr0 : r ≠ 0 := fun hr0 => by subst hr0; simp [hq] at hr
    have hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 := by simp [show _ = _ from hmp]
    have hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0 := hpn1.symm ▸ zn0.symm
    have hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0 := by
      simp only [leadingCoeff_pow' hpn0', leadingCoeff_eq_zero, hpn1, one_pow, one_mul, Ne.def,
          hr0, not_false_eq_true]
    

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
r : R[X]
hr : q = p ^ (natDegree q + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1
hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0
hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0
⊢ False

Next tactic:
have hnp : 0 < natDegree p := Nat.cast_lt.1 <| by
      rw [← degree_eq_natDegree hp0]; exact hp
theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : multiplicity.Finite p q :=
  have zn0 : (0 : R) ≠ 1 :=
    haveI := Nontrivial.of_polynomial_ne hq
    zero_ne_one
  ⟨natDegree q, fun ⟨r, hr⟩ => by
    have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at hp
    have hr0 : r ≠ 0 := fun hr0 => by subst hr0; simp [hq] at hr
    have hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 := by simp [show _ = _ from hmp]
    have hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0 := hpn1.symm ▸ zn0.symm
    have hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0 := by
      simp only [leadingCoeff_pow' hpn0', leadingCoeff_eq_zero, hpn1, one_pow, one_mul, Ne.def,
          hr0, not_false_eq_true]
    have hnp : 0 < natDegree p := Nat.cast_lt.1 <| by
      

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
r : R[X]
hr : q = p ^ (natDegree q + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1
hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0
hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0
⊢ ↑0 < ↑(natDegree p)

Next tactic:
rw [← degree_eq_natDegree hp0]
theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : multiplicity.Finite p q :=
  have zn0 : (0 : R) ≠ 1 :=
    haveI := Nontrivial.of_polynomial_ne hq
    zero_ne_one
  ⟨natDegree q, fun ⟨r, hr⟩ => by
    have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at hp
    have hr0 : r ≠ 0 := fun hr0 => by subst hr0; simp [hq] at hr
    have hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 := by simp [show _ = _ from hmp]
    have hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0 := hpn1.symm ▸ zn0.symm
    have hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0 := by
      simp only [leadingCoeff_pow' hpn0', leadingCoeff_eq_zero, hpn1, one_pow, one_mul, Ne.def,
          hr0, not_false_eq_true]
    have hnp : 0 < natDegree p := Nat.cast_lt.1 <| by
      rw [← degree_eq_natDegree hp0]; 

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
r : R[X]
hr : q = p ^ (natDegree q + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1
hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0
hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0
⊢ ↑0 < degree p
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
r : R[X]
hr : q = p ^ (natDegree q + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1
hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0
hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0
⊢ PartialOrder (WithBot ℕ)

Next tactic:
exact hp
theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : multiplicity.Finite p q :=
  have zn0 : (0 : R) ≠ 1 :=
    haveI := Nontrivial.of_polynomial_ne hq
    zero_ne_one
  ⟨natDegree q, fun ⟨r, hr⟩ => by
    have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at hp
    have hr0 : r ≠ 0 := fun hr0 => by subst hr0; simp [hq] at hr
    have hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 := by simp [show _ = _ from hmp]
    have hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0 := hpn1.symm ▸ zn0.symm
    have hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0 := by
      simp only [leadingCoeff_pow' hpn0', leadingCoeff_eq_zero, hpn1, one_pow, one_mul, Ne.def,
          hr0, not_false_eq_true]
    have hnp : 0 < natDegree p := Nat.cast_lt.1 <| by
      rw [← degree_eq_natDegree hp0]; exact hp
    

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
r : R[X]
hr : q = p ^ (natDegree q + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1
hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0
hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0
hnp : 0 < natDegree p
⊢ False

Next tactic:
have := congr_arg natDegree hr
theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : multiplicity.Finite p q :=
  have zn0 : (0 : R) ≠ 1 :=
    haveI := Nontrivial.of_polynomial_ne hq
    zero_ne_one
  ⟨natDegree q, fun ⟨r, hr⟩ => by
    have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at hp
    have hr0 : r ≠ 0 := fun hr0 => by subst hr0; simp [hq] at hr
    have hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 := by simp [show _ = _ from hmp]
    have hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0 := hpn1.symm ▸ zn0.symm
    have hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0 := by
      simp only [leadingCoeff_pow' hpn0', leadingCoeff_eq_zero, hpn1, one_pow, one_mul, Ne.def,
          hr0, not_false_eq_true]
    have hnp : 0 < natDegree p := Nat.cast_lt.1 <| by
      rw [← degree_eq_natDegree hp0]; exact hp
    have := congr_arg natDegree hr
    

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
r : R[X]
hr : q = p ^ (natDegree q + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1
hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0
hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0
hnp : 0 < natDegree p
this : natDegree q = natDegree (p ^ (natDegree q + 1) * r)
⊢ False

Next tactic:
rw [natDegree_mul' hpnr0, natDegree_pow' hpn0', add_mul, add_assoc] at this
theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : multiplicity.Finite p q :=
  have zn0 : (0 : R) ≠ 1 :=
    haveI := Nontrivial.of_polynomial_ne hq
    zero_ne_one
  ⟨natDegree q, fun ⟨r, hr⟩ => by
    have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at hp
    have hr0 : r ≠ 0 := fun hr0 => by subst hr0; simp [hq] at hr
    have hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 := by simp [show _ = _ from hmp]
    have hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0 := hpn1.symm ▸ zn0.symm
    have hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0 := by
      simp only [leadingCoeff_pow' hpn0', leadingCoeff_eq_zero, hpn1, one_pow, one_mul, Ne.def,
          hr0, not_false_eq_true]
    have hnp : 0 < natDegree p := Nat.cast_lt.1 <| by
      rw [← degree_eq_natDegree hp0]; exact hp
    have := congr_arg natDegree hr
    rw [natDegree_mul' hpnr0, natDegree_pow' hpn0', add_mul, add_assoc] at this
    

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
r : R[X]
hr : q = p ^ (natDegree q + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1
hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0
hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0
hnp : 0 < natDegree p
this : natDegree q = natDegree q * natDegree p + (1 * natDegree p + natDegree r)
⊢ False

Next tactic:
exact
      ne_of_lt
        (lt_add_of_le_of_pos (le_mul_of_one_le_right (Nat.zero_le _) hnp)
          (add_pos_of_pos_of_nonneg (by rwa [one_mul]) (Nat.zero_le _)))
        this
theorem multiplicity_finite_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : multiplicity.Finite p q :=
  have zn0 : (0 : R) ≠ 1 :=
    haveI := Nontrivial.of_polynomial_ne hq
    zero_ne_one
  ⟨natDegree q, fun ⟨r, hr⟩ => by
    have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at hp
    have hr0 : r ≠ 0 := fun hr0 => by subst hr0; simp [hq] at hr
    have hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 := by simp [show _ = _ from hmp]
    have hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0 := hpn1.symm ▸ zn0.symm
    have hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0 := by
      simp only [leadingCoeff_pow' hpn0', leadingCoeff_eq_zero, hpn1, one_pow, one_mul, Ne.def,
          hr0, not_false_eq_true]
    have hnp : 0 < natDegree p := Nat.cast_lt.1 <| by
      rw [← degree_eq_natDegree hp0]; exact hp
    have := congr_arg natDegree hr
    rw [natDegree_mul' hpnr0, natDegree_pow' hpn0', add_mul, add_assoc] at this
    exact
      ne_of_lt
        (lt_add_of_le_of_pos (le_mul_of_one_le_right (Nat.zero_le _) hnp)
          (add_pos_of_pos_of_nonneg (by 

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
hp : 0 < degree p
hmp : Monic p
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (natDegree q + 1) ∣ q
r : R[X]
hr : q = p ^ (natDegree q + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1
hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0
hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r ≠ 0
hnp : 0 < natDegree p
this : natDegree q = natDegree q * natDegree p + (1 * natDegree p + natDegree r)
⊢ 0 < 1 * natDegree p

Next tactic:
rwa [one_mul]
theorem div_wf_lemma (h : degree q ≤ degree p ∧ p ≠ 0) (hq : Monic q) :
    degree (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) < degree p :=
  have hp : leadingCoeff p ≠ 0 := mt leadingCoeff_eq_zero.1 h.2
  have hq0 : q ≠ 0 := hq.ne_zero_of_polynomial_ne h.2
  have hlt : natDegree q ≤ natDegree p :=
    Nat.cast_le.1
      (by 

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Ring R
p q : R[X]
h : degree q ≤ degree p ∧ p ≠ 0
hq : Monic q
hp : leadingCoeff p ≠ 0
hq0 : q ≠ 0
⊢ ↑(natDegree q) ≤ ↑(natDegree p)

Next tactic:
rw [← degree_eq_natDegree h.2, ← degree_eq_natDegree hq0]
theorem div_wf_lemma (h : degree q ≤ degree p ∧ p ≠ 0) (hq : Monic q) :
    degree (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) < degree p :=
  have hp : leadingCoeff p ≠ 0 := mt leadingCoeff_eq_zero.1 h.2
  have hq0 : q ≠ 0 := hq.ne_zero_of_polynomial_ne h.2
  have hlt : natDegree q ≤ natDegree p :=
    Nat.cast_le.1
      (by rw [← degree_eq_natDegree h.2, ← degree_eq_natDegree hq0]; 

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Ring R
p q : R[X]
h : degree q ≤ degree p ∧ p ≠ 0
hq : Monic q
hp : leadingCoeff p ≠ 0
hq0 : q ≠ 0
⊢ degree q ≤ degree p
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Ring R
p q : R[X]
⊢ degree q ≤ degree p ∧ p ≠ 0 → Monic q → leadingCoeff p ≠ 0 → q ≠ 0 → PartialOrder (WithBot ℕ)
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Ring R
p q : R[X]
⊢ degree q ≤ degree p ∧ p ≠ 0 → Monic q → leadingCoeff p ≠ 0 → q ≠ 0 → PartialOrder (WithBot ℕ)

Next tactic:
exact h.1
theorem div_wf_lemma (h : degree q ≤ degree p ∧ p ≠ 0) (hq : Monic q) :
    degree (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) < degree p :=
  have hp : leadingCoeff p ≠ 0 := mt leadingCoeff_eq_zero.1 h.2
  have hq0 : q ≠ 0 := hq.ne_zero_of_polynomial_ne h.2
  have hlt : natDegree q ≤ natDegree p :=
    Nat.cast_le.1
      (by rw [← degree_eq_natDegree h.2, ← degree_eq_natDegree hq0]; exact h.1)
  degree_sub_lt
    (by
      

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Ring R
p q : R[X]
h : degree q ≤ degree p ∧ p ≠ 0
hq : Monic q
hp : leadingCoeff p ≠ 0
hq0 : q ≠ 0
hlt : natDegree q ≤ natDegree p
⊢ degree p = degree (q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q)))

Next tactic:
rw [hq.degree_mul_comm, hq.degree_mul, degree_C_mul_X_pow _ hp, degree_eq_natDegree h.2,
        degree_eq_natDegree hq0, ← Nat.cast_add, tsub_add_cancel_of_le hlt]
theorem div_wf_lemma (h : degree q ≤ degree p ∧ p ≠ 0) (hq : Monic q) :
    degree (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) < degree p :=
  have hp : leadingCoeff p ≠ 0 := mt leadingCoeff_eq_zero.1 h.2
  have hq0 : q ≠ 0 := hq.ne_zero_of_polynomial_ne h.2
  have hlt : natDegree q ≤ natDegree p :=
    Nat.cast_le.1
      (by rw [← degree_eq_natDegree h.2, ← degree_eq_natDegree hq0]; exact h.1)
  degree_sub_lt
    (by
      rw [hq.degree_mul_comm, hq.degree_mul, degree_C_mul_X_pow _ hp, degree_eq_natDegree h.2,
        degree_eq_natDegree hq0, ← Nat.cast_add, tsub_add_cancel_of_le hlt])
    h.2 (by 

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Ring R
p q : R[X]
h : degree q ≤ degree p ∧ p ≠ 0
hq : Monic q
hp : leadingCoeff p ≠ 0
hq0 : q ≠ 0
hlt : natDegree q ≤ natDegree p
⊢ leadingCoeff p = leadingCoeff (q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q)))

Next tactic:
rw [leadingCoeff_monic_mul hq, leadingCoeff_mul_X_pow, leadingCoeff_C]
theorem degree_modByMonic_lt [Nontrivial R] :
    ∀ (p : R[X]) {q : R[X]} (_hq : Monic q), degree (p %ₘ q) < degree q
  | p, q, hq =>
    letI := Classical.decEq R
    if h : degree q ≤ degree p ∧ p ≠ 0 then by
      

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝¹ : Ring R
p✝ q✝ : R[X]
inst✝ : Nontrivial R
p q : R[X]
hq : Monic q
this : DecidableEq R := Classical.decEq R
h : degree q ≤ degree p ∧ p ≠ 0
⊢ degree (p %ₘ q) < degree q

Next tactic:
have _wf := div_wf_lemma ⟨h.1, h.2⟩ hq
theorem degree_modByMonic_lt [Nontrivial R] :
    ∀ (p : R[X]) {q : R[X]} (_hq : Monic q), degree (p %ₘ q) < degree q
  | p, q, hq =>
    letI := Classical.decEq R
    if h : degree q ≤ degree p ∧ p ≠ 0 then by
      have _wf := div_wf_lemma ⟨h.1, h.2⟩ hq
      

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝¹ : Ring R
p✝ q✝ : R[X]
inst✝ : Nontrivial R
p q : R[X]
hq : Monic q
this : DecidableEq R := Classical.decEq R
h : degree q ≤ degree p ∧ p ≠ 0
_wf : degree (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) < degree p
⊢ degree (p %ₘ q) < degree q

Next tactic:
have :=
        degree_modByMonic_lt (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) hq
theorem degree_modByMonic_lt [Nontrivial R] :
    ∀ (p : R[X]) {q : R[X]} (_hq : Monic q), degree (p %ₘ q) < degree q
  | p, q, hq =>
    letI := Classical.decEq R
    if h : degree q ≤ degree p ∧ p ≠ 0 then by
      have _wf := div_wf_lemma ⟨h.1, h.2⟩ hq
      have :=
        degree_modByMonic_lt (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) hq
      

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝¹ : Ring R
p✝ q✝ : R[X]
inst✝ : Nontrivial R
p q : R[X]
hq : Monic q
this✝ : DecidableEq R := Classical.decEq R
h : degree q ≤ degree p ∧ p ≠ 0
_wf : degree (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) < degree p
this : degree ((p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) %ₘ q) < degree q
⊢ degree (p %ₘ q) < degree q

Next tactic:
unfold modByMonic at this ⊢
theorem degree_modByMonic_lt [Nontrivial R] :
    ∀ (p : R[X]) {q : R[X]} (_hq : Monic q), degree (p %ₘ q) < degree q
  | p, q, hq =>
    letI := Classical.decEq R
    if h : degree q ≤ degree p ∧ p ≠ 0 then by
      have _wf := div_wf_lemma ⟨h.1, h.2⟩ hq
      have :=
        degree_modByMonic_lt (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) hq
      unfold modByMonic at this ⊢
      

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝¹ : Ring R
p✝ q✝ : R[X]
inst✝ : Nontrivial R
p q : R[X]
hq : Monic q
this✝ : DecidableEq R := Classical.decEq R
h : degree q ≤ degree p ∧ p ≠ 0
_wf : degree (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) < degree p
this :
  degree
      (if hq : Monic q then (divModByMonicAux (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) hq).2
      else p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) <
    degree q
⊢ degree (if hq : Monic q then (divModByMonicAux p hq).2 else p) < degree q

Next tactic:
unfold divModByMonicAux
theorem degree_modByMonic_lt [Nontrivial R] :
    ∀ (p : R[X]) {q : R[X]} (_hq : Monic q), degree (p %ₘ q) < degree q
  | p, q, hq =>
    letI := Classical.decEq R
    if h : degree q ≤ degree p ∧ p ≠ 0 then by
      have _wf := div_wf_lemma ⟨h.1, h.2⟩ hq
      have :=
        degree_modByMonic_lt (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) hq
      unfold modByMonic at this ⊢
      unfold divModByMonicAux
      

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝¹ : Ring R
p✝ q✝ : R[X]
inst✝ : Nontrivial R
p q : R[X]
hq : Monic q
this✝ : DecidableEq R := Classical.decEq R
h : degree q ≤ degree p ∧ p ≠ 0
_wf : degree (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) < degree p
this :
  degree
      (if hq : Monic q then (divModByMonicAux (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) hq).2
      else p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) <
    degree q
⊢ degree
      (if h : Monic q then
        (if h_1 : degree q ≤ degree p ∧ p ≠ 0 then
            let z := C (leadingCoeff p) * X ^ (natDegree p - natDegree q);
            let_fun _wf := (_ : degree (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) < degree p);
            let dm := divModByMonicAux (p - q * z) (_ : Monic q);
            (z + dm.1, dm.2)
          else (0, p)).2
      else p) <
    degree q

Next tactic:
dsimp
theorem degree_modByMonic_lt [Nontrivial R] :
    ∀ (p : R[X]) {q : R[X]} (_hq : Monic q), degree (p %ₘ q) < degree q
  | p, q, hq =>
    letI := Classical.decEq R
    if h : degree q ≤ degree p ∧ p ≠ 0 then by
      have _wf := div_wf_lemma ⟨h.1, h.2⟩ hq
      have :=
        degree_modByMonic_lt (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) hq
      unfold modByMonic at this ⊢
      unfold divModByMonicAux
      dsimp
      

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝¹ : Ring R
p✝ q✝ : R[X]
inst✝ : Nontrivial R
p q : R[X]
hq : Monic q
this✝ : DecidableEq R := Classical.decEq R
h : degree q ≤ degree p ∧ p ≠ 0
_wf : degree (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) < degree p
this :
  degree
      (if hq : Monic q then (divModByMonicAux (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) hq).2
      else p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) <
    degree q
⊢ degree
      (if h : Monic q then
        (if degree q ≤ degree p ∧ ¬p = 0 then
            (C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +
                (divModByMonicAux (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) (_ : Monic q)).1,
              (divModByMonicAux (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) (_ : Monic q)).2)
          else (0, p)).2
      else p) <
    degree q

Next tactic:
rw [dif_pos hq] at this ⊢
theorem degree_modByMonic_lt [Nontrivial R] :
    ∀ (p : R[X]) {q : R[X]} (_hq : Monic q), degree (p %ₘ q) < degree q
  | p, q, hq =>
    letI := Classical.decEq R
    if h : degree q ≤ degree p ∧ p ≠ 0 then by
      have _wf := div_wf_lemma ⟨h.1, h.2⟩ hq
      have :=
        degree_modByMonic_lt (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) hq
      unfold modByMonic at this ⊢
      unfold divModByMonicAux
      dsimp
      rw [dif_pos hq] at this ⊢
      

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝¹ : Ring R
p✝ q✝ : R[X]
inst✝ : Nontrivial R
p q : R[X]
hq : Monic q
this✝ : DecidableEq R := Classical.decEq R
h : degree q ≤ degree p ∧ p ≠ 0
_wf : degree (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) < degree p
this : degree (divModByMonicAux (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) hq).2 < degree q
⊢ degree
      (if degree q ≤ degree p ∧ ¬p = 0 then
          (C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +
              (divModByMonicAux (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) (_ : Monic q)).1,
            (divModByMonicAux (p - q * (C (leadingCoeff p) * X ^ (natDegree p - natDegree q))) (_ : Monic q)).2)
        else (0, p)).2 <
    degree q

Next tactic:
rw [if_pos h]