-- Mathlib/Algebra/Polynomial/UnitTrinomial.lean
theorem irreducible_of_coprime (hp : p.IsUnitTrinomial)
    (h : IsRelPrime p p.mirror) : Irreducible p

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p : ℤ[X]
h : IsRelPrime p p.mirror
q : ℤ[X]
hpq : p * p.mirror = q * q.mirror
k m n : ℕ
hkm : k < m
hmn : m < n
u v w : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' ↑x ↑y ↑z
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror

Tactic:
have hk : k = k' := by
  rw [← mul_right_inj' (show 2 ≠ 0 from two_ne_zero), ←
    trinomial_natTrailingDegree hkm hmn u.ne_zero, ← hp, ← natTrailingDegree_mul_mirror, hpq,
    natTrailingDegree_mul_mirror, hq, trinomial_natTrailingDegree hkm' hmn' x.ne_zero]

Result:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p : ℤ[X]
h : IsRelPrime p p.mirror
q : ℤ[X]
hpq : p * p.mirror = q * q.mirror
k m n : ℕ
hkm : k < m
hmn : m < n
u v w : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' ↑x ↑y ↑z
hk : k = k'
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror

Proof state:
p : ℤ[X]
hp : p.IsUnitTrinomial
h : IsRelPrime p p.mirror
⊢ Irreducible p

Tactic:
refine irreducible_of_mirror hp.not_isUnit (fun q hpq => ?_) h

Result:
p : ℤ[X]
hp : p.IsUnitTrinomial
h : IsRelPrime p p.mirror
q : ℤ[X]
hpq : p * p.mirror = q * q.mirror
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro
p : ℤ[X]
h : IsRelPrime p p.mirror
q : ℤ[X]
hpq : p * p.mirror = q * q.mirror
hq : q.IsUnitTrinomial
k m n : ℕ
hkm : k < m
hmn : m < n
u v w : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror

Tactic:
obtain ⟨k', m', n', hkm', hmn', x, y, z, hq⟩ := hq

Result:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p : ℤ[X]
h : IsRelPrime p p.mirror
q : ℤ[X]
hpq : p * p.mirror = q * q.mirror
k m n : ℕ
hkm : k < m
hmn : m < n
u v w : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' ↑x ↑y ↑z
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr
p : ℤ[X]
h : IsRelPrime p p.mirror
q : ℤ[X]
hpq : -p * (-p).mirror = q * q.mirror
k m n : ℕ
hkm : k < m
hmn : m < n
u v w : ℤˣ
hp : -p = C (-↑u) * X ^ k + C (-↑v) * X ^ m + C (-↑w) * X ^ n
m' : ℕ
x y z : ℤˣ
hkm' : k < m'
hmn' : m' < n
hq : q = trinomial k m' n (↑x) (-↑v) ↑z
h1 : ↑y = -↑v
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror

Tactic:
rcases irreducible_aux3 hkm hmn hkm' hmn' (-u) (-v) (-w) x z hp hq hpq with (rfl | rfl)

Result:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.inl
p : ℤ[X]
h : IsRelPrime p p.mirror
k m n : ℕ
hkm : k < m
hmn : m < n
u v w : ℤˣ
hp : -p = C (-↑u) * X ^ k + C (-↑v) * X ^ m + C (-↑w) * X ^ n
m' : ℕ
x y z : ℤˣ
hkm' : k < m'
hmn' : m' < n
h1 : ↑y = -↑v
hpq : -p * (-p).mirror = -p * (-p).mirror
hq : -p = trinomial k m' n (↑x) (-↑v) ↑z
⊢ -p = p ∨ -p = -p ∨ -p = p.mirror ∨ -p = -p.mirror

case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.inr
p : ℤ[X]
h : IsRelPrime p p.mirror
k m n : ℕ
hkm : k < m
hmn : m < n
u v w : ℤˣ
hp : -p = C (-↑u) * X ^ k + C (-↑v) * X ^ m + C (-↑w) * X ^ n
m' : ℕ
x y z : ℤˣ
hkm' : k < m'
hmn' : m' < n
h1 : ↑y = -↑v
hpq : -p * (-p).mirror = (-p).mirror * (-p).mirror.mirror
hq : (-p).mirror = trinomial k m' n (↑x) (-↑v) ↑z
⊢ (-p).mirror = p ∨ (-p).mirror = -p ∨ (-p).mirror = p.mirror ∨ (-p).mirror = -p.mirror