-- Mathlib/Algebra/Polynomial/Div.lean
theorem finiteMultiplicity_of_degree_pos_of_monic (hp : (0 : WithBot ℕ) < degree p) (hmp : Monic p)
    (hq : q ≠ 0) : FiniteMultiplicity 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

Proof state:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
⊢ p.leadingCoeff ^ (q.natDegree + 1) = 1

Tactic:
simp [show _ = _ from hmp]

Result:
no goals

Proof state:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
⊢ False

Tactic:
have hp0 : p ≠ 0 := fun hp0 => by simp [hp0] at hp

Result:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p ≠ 0
⊢ False

Proof state:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p ≠ 0
hr0 : r = 0
⊢ False

Tactic:
subst hr0

Result:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
hp0 : p ≠ 0
hr : q = p ^ (q.natDegree + 1) * 0
⊢ False

Proof state:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : p.leadingCoeff ^ (q.natDegree + 1) = 1
hpn0' : p.leadingCoeff ^ (q.natDegree + 1) ≠ 0
hpnr0 : (p ^ (q.natDegree + 1)).leadingCoeff * r.leadingCoeff ≠ 0
⊢ ↑0 < p.degree

Tactic:
exact hp

Result:
no goals

Proof state:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : p.leadingCoeff ^ (q.natDegree + 1) = 1
hpn0' : p.leadingCoeff ^ (q.natDegree + 1) ≠ 0
hpnr0 : (p ^ (q.natDegree + 1)).leadingCoeff * r.leadingCoeff ≠ 0
hnp : 0 < p.natDegree
⊢ False

Tactic:
have := congr_arg natDegree hr

Result:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : p.leadingCoeff ^ (q.natDegree + 1) = 1
hpn0' : p.leadingCoeff ^ (q.natDegree + 1) ≠ 0
hpnr0 : (p ^ (q.natDegree + 1)).leadingCoeff * r.leadingCoeff ≠ 0
hnp : 0 < p.natDegree
this : q.natDegree = (p ^ (q.natDegree + 1) * r).natDegree
⊢ False

Proof state:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : p.leadingCoeff ^ (q.natDegree + 1) = 1
hpn0' : p.leadingCoeff ^ (q.natDegree + 1) ≠ 0
hpnr0 : (p ^ (q.natDegree + 1)).leadingCoeff * r.leadingCoeff ≠ 0
hnp : 0 < p.natDegree
this : q.natDegree = q.natDegree * p.natDegree + (1 * p.natDegree + r.natDegree)
⊢ 0 < 1 * p.natDegree

Tactic:
rwa [one_mul]

Result:
no goals

Proof state:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : p.leadingCoeff ^ (q.natDegree + 1) = 1
hpn0' : p.leadingCoeff ^ (q.natDegree + 1) ≠ 0
hpnr0 : (p ^ (q.natDegree + 1)).leadingCoeff * r.leadingCoeff ≠ 0
⊢ ↑0 < ↑p.natDegree

Tactic:
rw [← degree_eq_natDegree hp0]

Result:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : p.leadingCoeff ^ (q.natDegree + 1) = 1
hpn0' : p.leadingCoeff ^ (q.natDegree + 1) ≠ 0
hpnr0 : (p ^ (q.natDegree + 1)).leadingCoeff * r.leadingCoeff ≠ 0
⊢ ↑0 < p.degree

Proof state:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : p.leadingCoeff ^ (q.natDegree + 1) = 1
⊢ False

Tactic:
have hpn0' : leadingCoeff p ^ (natDegree q + 1) ≠ 0 := hpn1.symm ▸ zn0.symm

Result:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : p.leadingCoeff ^ (q.natDegree + 1) = 1
hpn0' : p.leadingCoeff ^ (q.natDegree + 1) ≠ 0
⊢ False

Proof state:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : p.leadingCoeff ^ (q.natDegree + 1) = 1
hpn0' : p.leadingCoeff ^ (q.natDegree + 1) ≠ 0
hpnr0 : (p ^ (q.natDegree + 1)).leadingCoeff * r.leadingCoeff ≠ 0
hnp : 0 < p.natDegree
this : q.natDegree = q.natDegree * p.natDegree + (1 * p.natDegree + r.natDegree)
⊢ False

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

Result:
no goals

Proof state:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p = 0
⊢ False

Tactic:
simp [hp0] at hp

Result:
no goals

Proof state:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : p.leadingCoeff ^ (q.natDegree + 1) = 1
hpn0' : p.leadingCoeff ^ (q.natDegree + 1) ≠ 0
hpnr0 : (p ^ (q.natDegree + 1)).leadingCoeff * r.leadingCoeff ≠ 0
hnp : 0 < p.natDegree
this : q.natDegree = (p ^ (q.natDegree + 1) * r).natDegree
⊢ False

Tactic:
rw [natDegree_mul' hpnr0, natDegree_pow' hpn0', add_mul, add_assoc] at this

Result:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : p.leadingCoeff ^ (q.natDegree + 1) = 1
hpn0' : p.leadingCoeff ^ (q.natDegree + 1) ≠ 0
hpnr0 : (p ^ (q.natDegree + 1)).leadingCoeff * r.leadingCoeff ≠ 0
hnp : 0 < p.natDegree
this : q.natDegree = q.natDegree * p.natDegree + (1 * p.natDegree + r.natDegree)
⊢ False

Proof state:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : p.leadingCoeff ^ (q.natDegree + 1) = 1
hpn0' : p.leadingCoeff ^ (q.natDegree + 1) ≠ 0
⊢ (p ^ (q.natDegree + 1)).leadingCoeff * r.leadingCoeff ≠ 0

Tactic:
simp only [leadingCoeff_pow' hpn0', leadingCoeff_eq_zero, hpn1, one_pow, one_mul, Ne,
    hr0, not_false_eq_true]

Result:
no goals

Proof state:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
⊢ False

Tactic:
have hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 := by simp [show _ = _ from hmp]

Result:
R : Type u
inst✝ : Semiring R
p q : R[X]
hp : 0 < p.degree
hmp : p.Monic
hq : q ≠ 0
zn0 : 0 ≠ 1
x✝ : p ^ (q.natDegree + 1) ∣ q
r : R[X]
hr : q = p ^ (q.natDegree + 1) * r
hp0 : p ≠ 0
hr0 : r ≠ 0
hpn1 : p.leadingCoeff ^ (q.natDegree + 1) = 1
⊢ False