Theorem: Polynomial.irreducible_of_eisenstein_criterion
File: Mathlib/RingTheory/EisensteinCriterion.lean

Proof steps:
state_before: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
f : R[X]
P : Ideal R
hP : P.IsPrime
hfl : f.leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < f.degree → f.coeff n ∈ P
hfd0 : 0 < f.degree
h0 : f.coeff 0 ∉ P ^ 2
hu : f.IsPrimitive
x✝ : f = 0
⊢ False
state_after: no goals
tactic: simp_all only [not_true, Submodule.zero_mem, coeff_zero]
state_before: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
f : R[X]
P : Ideal R
hP : P.IsPrime
hfl : f.leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < f.degree → f.coeff n ∈ P
hfd0✝ : 0 < f.degree
h0 : f.coeff 0 ∉ P ^ 2
hu : f.IsPrimitive
hf0 : f ≠ 0
hf : map (mk P) f = C ((mk P) f.leadingCoeff) * X ^ f.natDegree
hfd0 : 0 < f.natDegree
h : f.degree = 0
⊢ False
state_after: no goals
tactic: simp_all only [lt_irrefl]
state_before: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
f : R[X]
P : Ideal R
hP : P.IsPrime
hfl : f.leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < f.degree → f.coeff n ∈ P
hfd0✝ : 0 < f.degree
h0 : f.coeff 0 ∉ P ^ 2
hu : f.IsPrimitive
hf0 : f ≠ 0
hf : map (mk P) f = C ((mk P) f.leadingCoeff) * X ^ f.natDegree
hfd0 : 0 < f.natDegree
⊢ ∀ (a b : R[X]), f = a * b → IsUnit a ∨ IsUnit b
state_after: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) (p * q) = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
⊢ IsUnit p ∨ IsUnit q
tactic: rintro p q rfl
state_before: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) (p * q) = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
⊢ IsUnit p ∨ IsUnit q
state_after: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
⊢ IsUnit p ∨ IsUnit q
tactic: rw [Polynomial.map_mul] at hf
state_before: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
⊢ IsUnit p ∨ IsUnit q
state_after: case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
m n : ℕ
b c : (R ⧸ P)[X]
hmnd : m + n = (p * q).natDegree
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hp : map (mk P) p = b * X ^ m
hq : map (mk P) q = c * X ^ n
⊢ IsUnit p ∨ IsUnit q
tactic: rcases mul_eq_mul_prime_pow
        (show Prime (X : Polynomial (R ⧸ P)) from monic_X.prime_of_degree_eq_one degree_X) hf with
      ⟨m, n, b, c, hmnd, hbc, hp, hq⟩
state_before: case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
m n : ℕ
b c : (R ⧸ P)[X]
hmnd : m + n = (p * q).natDegree
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hp : map (mk P) p = b * X ^ m
hq : map (mk P) q = c * X ^ n
⊢ IsUnit p ∨ IsUnit q
state_after: case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
m n : ℕ
b c : (R ⧸ P)[X]
hmnd : m + n = (p * q).natDegree
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hp : map (mk P) p = b * X ^ m
hq : map (mk P) q = c * X ^ n
hmn : 0 < m → 0 < n → False
⊢ IsUnit p ∨ IsUnit q
tactic: have hmn : 0 < m → 0 < n → False := by
      intro hm0 hn0
      refine h0 ?_
      rw [coeff_zero_eq_eval_zero, eval_mul, sq]
      exact
        Ideal.mul_mem_mul (eval_zero_mem_ideal_of_eq_mul_X_pow hp hm0.ne')
          (eval_zero_mem_ideal_of_eq_mul_X_pow hq hn0.ne')
state_before: case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
m n : ℕ
b c : (R ⧸ P)[X]
hmnd : m + n = (p * q).natDegree
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hp : map (mk P) p = b * X ^ m
hq : map (mk P) q = c * X ^ n
hmn : 0 < m → 0 < n → False
⊢ IsUnit p ∨ IsUnit q
state_after: case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
m n : ℕ
b c : (R ⧸ P)[X]
hmnd : m + n = (p * q).natDegree
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hp : map (mk P) p = b * X ^ m
hq : map (mk P) q = c * X ^ n
hmn : 0 < m → 0 < n → False
hpql0 : (mk P) (p * q).leadingCoeff ≠ 0
⊢ IsUnit p ∨ IsUnit q
tactic: have hpql0 : (mk P) (p * q).leadingCoeff ≠ 0 := by rwa [Ne, eq_zero_iff_mem]
state_before: case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
m n : ℕ
b c : (R ⧸ P)[X]
hmnd : m + n = (p * q).natDegree
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hp : map (mk P) p = b * X ^ m
hq : map (mk P) q = c * X ^ n
hmn : 0 < m → 0 < n → False
hpql0 : (mk P) (p * q).leadingCoeff ≠ 0
⊢ IsUnit p ∨ IsUnit q
state_after: case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
m n : ℕ
b c : (R ⧸ P)[X]
hmnd : m + n = (p * q).natDegree
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hp : map (mk P) p = b * X ^ m
hq : map (mk P) q = c * X ^ n
hmn : 0 < m → 0 < n → False
hpql0 : (mk P) (p * q).leadingCoeff ≠ 0
hp0 : p ≠ 0
⊢ IsUnit p ∨ IsUnit q
tactic: have hp0 : p ≠ 0 := fun h => by
      simp_all only [zero_mul, eq_self_iff_true, not_true, Ne]
state_before: case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
m n : ℕ
b c : (R ⧸ P)[X]
hmnd : m + n = (p * q).natDegree
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hp : map (mk P) p = b * X ^ m
hq : map (mk P) q = c * X ^ n
hmn : 0 < m → 0 < n → False
hpql0 : (mk P) (p * q).leadingCoeff ≠ 0
hp0 : p ≠ 0
⊢ IsUnit p ∨ IsUnit q
state_after: case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
m n : ℕ
b c : (R ⧸ P)[X]
hmnd : m + n = (p * q).natDegree
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hp : map (mk P) p = b * X ^ m
hq : map (mk P) q = c * X ^ n
hmn : 0 < m → 0 < n → False
hpql0 : (mk P) (p * q).leadingCoeff ≠ 0
hp0 : p ≠ 0
hq0 : q ≠ 0
⊢ IsUnit p ∨ IsUnit q
tactic: have hq0 : q ≠ 0 := fun h => by
      simp_all only [eq_self_iff_true, not_true, Ne, mul_zero]
state_before: case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
m n : ℕ
b c : (R ⧸ P)[X]
hmnd : m + n = (p * q).natDegree
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hp : map (mk P) p = b * X ^ m
hq : map (mk P) q = c * X ^ n
hmn : 0 < m → 0 < n → False
hpql0 : (mk P) (p * q).leadingCoeff ≠ 0
hp0 : p ≠ 0
hq0 : q ≠ 0
⊢ IsUnit p ∨ IsUnit q
state_after: case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
m n : ℕ
b c : (R ⧸ P)[X]
hmnd : m + n = (p * q).natDegree
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hp : map (mk P) p = b * X ^ m
hq : map (mk P) q = c * X ^ n
hmn : 0 < m → 0 < n → False
hpql0 : (mk P) (p * q).leadingCoeff ≠ 0
hp0 : p ≠ 0
hq0 : q ≠ 0
hbc0 : b.degree = 0 ∧ c.degree = 0
⊢ IsUnit p ∨ IsUnit q
tactic: have hbc0 : degree b = 0 ∧ degree c = 0 := by
      apply_fun degree at hbc
      rwa [degree_C hpql0, degree_mul, eq_comm, Nat.WithBot.add_eq_zero_iff] at hbc
state_before: case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
m n : ℕ
b c : (R ⧸ P)[X]
hmnd : m + n = (p * q).natDegree
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hp : map (mk P) p = b * X ^ m
hq : map (mk P) q = c * X ^ n
hmn : 0 < m → 0 < n → False
hpql0 : (mk P) (p * q).leadingCoeff ≠ 0
hp0 : p ≠ 0
hq0 : q ≠ 0
hbc0 : b.degree = 0 ∧ c.degree = 0
⊢ IsUnit p ∨ IsUnit q
state_after: case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
m n : ℕ
b c : (R ⧸ P)[X]
hmnd : m + n = (p * q).natDegree
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hp : map (mk P) p = b * X ^ m
hq : map (mk P) q = c * X ^ n
hmn : 0 < m → 0 < n → False
hpql0 : (mk P) (p * q).leadingCoeff ≠ 0
hp0 : p ≠ 0
hq0 : q ≠ 0
hbc0 : b.degree = 0 ∧ c.degree = 0
hmp : m ≤ p.natDegree
⊢ IsUnit p ∨ IsUnit q
tactic: have hmp : m ≤ natDegree p := le_natDegree_of_map_eq_mul_X_pow hP hp hbc0.1
state_before: case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
m n : ℕ
b c : (R ⧸ P)[X]
hmnd : m + n = (p * q).natDegree
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hp : map (mk P) p = b * X ^ m
hq : map (mk P) q = c * X ^ n
hmn : 0 < m → 0 < n → False
hpql0 : (mk P) (p * q).leadingCoeff ≠ 0
hp0 : p ≠ 0
hq0 : q ≠ 0
hbc0 : b.degree = 0 ∧ c.degree = 0
hmp : m ≤ p.natDegree
⊢ IsUnit p ∨ IsUnit q
state_after: case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
m n : ℕ
b c : (R ⧸ P)[X]
hmnd : m + n = (p * q).natDegree
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hp : map (mk P) p = b * X ^ m
hq : map (mk P) q = c * X ^ n
hmn : 0 < m → 0 < n → False
hpql0 : (mk P) (p * q).leadingCoeff ≠ 0
hp0 : p ≠ 0
hq0 : q ≠ 0
hbc0 : b.degree = 0 ∧ c.degree = 0
hmp : m ≤ p.natDegree
hnq : n ≤ q.natDegree
⊢ IsUnit p ∨ IsUnit q
tactic: have hnq : n ≤ natDegree q := le_natDegree_of_map_eq_mul_X_pow hP hq hbc0.2
state_before: case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
m n : ℕ
b c : (R ⧸ P)[X]
hmnd : m + n = (p * q).natDegree
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hp : map (mk P) p = b * X ^ m
hq : map (mk P) q = c * X ^ n
hmn : 0 < m → 0 < n → False
hpql0 : (mk P) (p * q).leadingCoeff ≠ 0
hp0 : p ≠ 0
hq0 : q ≠ 0
hbc0 : b.degree = 0 ∧ c.degree = 0
hmp : m ≤ p.natDegree
hnq : n ≤ q.natDegree
hpmqn : p.natDegree = m ∧ q.natDegree = n
⊢ IsUnit p ∨ IsUnit q
state_after: case intro.intro.intro.intro.intro.intro.intro.inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
n : ℕ
b c : (R ⧸ P)[X]
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hq : map (mk P) q = c * X ^ n
hpql0 : (mk P) (p * q).leadingCoeff ≠ 0
hp0 : p ≠ 0
hq0 : q ≠ 0
hbc0 : b.degree = 0 ∧ c.degree = 0
hnq : n ≤ q.natDegree
hmnd : 0 + n = (p * q).natDegree
hp : map (mk P) p = b * X ^ 0
hmn : 0 < 0 → 0 < n → False
hmp : 0 ≤ p.natDegree
hpmqn : p.natDegree = 0 ∧ q.natDegree = n
⊢ IsUnit p ∨ IsUnit q

case intro.intro.intro.intro.intro.intro.intro.inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
P : Ideal R
hP : P.IsPrime
p q : R[X]
hfl : (p * q).leadingCoeff ∉ P
hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P
hfd0✝ : 0 < (p * q).degree
h0 : (p * q).coeff 0 ∉ P ^ 2
hu : (p * q).IsPrimitive
hf0 : p * q ≠ 0
hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree
hfd0 : 0 < (p * q).natDegree
m : ℕ
b c : (R ⧸ P)[X]
hbc : C ((mk P) (p * q).leadingCoeff) = b * c
hp : map (mk P) p = b * X ^ m
hpql0 : (mk P) (p * q).leadingCoeff ≠ 0
hp0 : p ≠ 0
hq0 : q ≠ 0
hbc0 : b.degree = 0 ∧ c.degree = 0
hmp : m ≤ p.natDegree
hmnd : m + 0 = (p * q).natDegree
hq : map (mk P) q = c * X ^ 0
hmn : 0 < m → 0 < 0 → False
hnq : 0 ≤ q.natDegree
hpmqn : p.natDegree = m ∧ q.natDegree = 0
⊢ IsUnit p ∨ IsUnit q
tactic: obtain rfl | rfl : m = 0 ∨ n = 0 := by
      rwa [pos_iff_ne_zero, pos_iff_ne_zero, imp_false, Classical.not_not, ← or_iff_not_imp_left]
        at hmn