Theorem: Polynomial.IsUnitTrinomial.irreducible_of_coprime
File: Mathlib/Algebra/Polynomial/UnitTrinomial.lean

Proof steps:
state_before: p q : ℤ[X]
hp : p.IsUnitTrinomial
h : IsRelPrime p p.mirror
⊢ Irreducible p
state_after: p q✝ : 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
tactic: refine irreducible_of_mirror hp.not_isUnit (fun q hpq => ?_) h
state_before: p q✝ : 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
state_after: p q✝ : ℤ[X]
hp : p.IsUnitTrinomial
h : IsRelPrime p p.mirror
q : ℤ[X]
hpq : p * p.mirror = q * q.mirror
hq : q.IsUnitTrinomial
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror
tactic: have hq : IsUnitTrinomial q := (isUnitTrinomial_iff'' hpq).mp hp
state_before: p q✝ : ℤ[X]
hp : p.IsUnitTrinomial
h : IsRelPrime p p.mirror
q : ℤ[X]
hpq : p * p.mirror = q * q.mirror
hq : q.IsUnitTrinomial
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror
state_after: case intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : 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, u, v, w, hp⟩ := hp
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : 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
state_after: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : 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: obtain ⟨k', m', n', hkm', hmn', x, y, z, hq⟩ := hq
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : 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
state_after: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : 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
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]
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : 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
state_after: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : 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'
hn : n = n'
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror
tactic: have hn : n = n' := by
    rw [← mul_right_inj' (show 2 ≠ 0 from two_ne_zero), ← trinomial_natDegree hkm hmn w.ne_zero, ←
      hp, ← natDegree_mul_mirror, hpq, natDegree_mul_mirror, hq,
      trinomial_natDegree hkm' hmn' z.ne_zero]
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : 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'
hn : n = n'
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror
state_after: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : 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
m' n' : ℕ
hmn' : m' < n'
x y z : ℤˣ
hn : n = n'
hkm' : k < m'
hq : q = trinomial k m' n' ↑x ↑y ↑z
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror
tactic: subst hk
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : 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
m' n' : ℕ
hmn' : m' < n'
x y z : ℤˣ
hn : n = n'
hkm' : k < m'
hq : q = trinomial k m' n' ↑x ↑y ↑z
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror
state_after: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : 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
m' : ℕ
x y z : ℤˣ
hkm' : k < m'
hmn' : m' < n
hq : q = trinomial k m' n ↑x ↑y ↑z
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror
tactic: subst hn
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : 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
m' : ℕ
x y z : ℤˣ
hkm' : k < m'
hmn' : m' < n
hq : q = trinomial k m' n ↑x ↑y ↑z
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror
state_after: case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inl
p q✝ : 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
m' : ℕ
x y z : ℤˣ
hkm' : k < m'
hmn' : m' < n
hq : q = trinomial k m' n ↑x ↑y ↑z
h1 : ↑y = ↑v
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror

case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr
p q✝ : 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
m' : ℕ
x y z : ℤˣ
hkm' : k < m'
hmn' : m' < n
hq : q = trinomial k m' n ↑x ↑y ↑z
h1 : ↑y = -↑v
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror
tactic: rcases eq_or_eq_neg_of_sq_eq_sq (y : ℤ) (v : ℤ)
      ((Int.isUnit_sq y.isUnit).trans (Int.isUnit_sq v.isUnit).symm) with
    (h1 | h1)