-- Mathlib/Algebra/Polynomial/UnitTrinomial.lean
theorem irreducible_aux3 {k m m' n : ℕ} (hkm : k < m) (hmn : m < n) (hkm' : k < m') (hmn' : m' < n)
    (u v w x z : Units ℤ) (hp : p = trinomial k m n (u : ℤ) v w)
    (hq : q = trinomial k m' n (x : ℤ) v z) (h : p * p.mirror = q * q.mirror) :
    q = p ∨ q = p.mirror

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

Tactic:
rw [mul_comm q, ← q.mirror_mirror, q.mirror.mirror_mirror] at h

Result:
case inr.intro
p q : ℤ[X]
k m m' n : ℕ
hkm : k < m
hmn : m < n
hkm' : k < m'
hmn' : m' < n
u v w : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
h : p * p.mirror = q.mirror * q.mirror.mirror
hq : q.mirror = trinomial k (n - m' + k) n ↑u ↑v ↑w
hmul : ↑w * ↑u = ↑u * ↑w
⊢ q = p ∨ q = p.mirror

Proof state:
p q : ℤ[X]
k m m' n : ℕ
hkm : k < m
hmn : m < n
hkm' : k < m'
hmn' : m' < n
u v w x z : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
hq : q = trinomial k m' n ↑x ↑v ↑z
h : p * p.mirror = q * q.mirror
hmul : ↑w * ↑u = ↑z * ↑x
hadd : ↑w + ↑u = ↑z + ↑x
⊢ q = p ∨ q = p.mirror

Tactic:
replace hadd :=
  (Int.isUnit_add_isUnit_eq_isUnit_add_isUnit w.isUnit u.isUnit z.isUnit x.isUnit).mp hadd

Result:
p q : ℤ[X]
k m m' n : ℕ
hkm : k < m
hmn : m < n
hkm' : k < m'
hmn' : m' < n
u v w x z : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
hq : q = trinomial k m' n ↑x ↑v ↑z
h : p * p.mirror = q * q.mirror
hmul : ↑w * ↑u = ↑z * ↑x
hadd : ↑w = ↑z ∧ ↑u = ↑x ∨ ↑w = ↑x ∧ ↑u = ↑z
⊢ q = p ∨ q = p.mirror

Proof state:
p q : ℤ[X]
k m m' n : ℕ
hkm : k < m
hmn : m < n
hkm' : k < m'
hmn' : m' < n
u v w x z : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
hq : q = trinomial k m' n ↑x ↑v ↑z
h : p * p.mirror = q * q.mirror
hmul : ↑w * ↑u = ↑z * ↑x
hadd : w = z ∧ u = x ∨ w = x ∧ u = z
⊢ q = p ∨ q = p.mirror

Tactic:
rcases hadd with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)

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

case inr.intro
p q : ℤ[X]
k m m' n : ℕ
hkm : k < m
hmn : m < n
hkm' : k < m'
hmn' : m' < n
u v w : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
h : p * p.mirror = q * q.mirror
hq : q = trinomial k m' n ↑w ↑v ↑u
hmul : ↑w * ↑u = ↑u * ↑w
⊢ q = p ∨ q = p.mirror

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

Tactic:
rw [← mirror_inj, or_comm, ← mirror_eq_iff]

Result:
case inr.intro
p q : ℤ[X]
k m m' n : ℕ
hkm : k < m
hmn : m < n
hkm' : k < m'
hmn' : m' < n
u v w : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
h : p * p.mirror = q.mirror * q.mirror.mirror
hq : q.mirror = trinomial k (n - m' + k) n ↑u ↑v ↑w
hmul : ↑w * ↑u = ↑u * ↑w
⊢ q.mirror = p ∨ q.mirror = p.mirror

Proof state:
p q : ℤ[X]
k m m' n : ℕ
hkm : k < m
hmn : m < n
hkm' : k < m'
hmn' : m' < n
u v w x z : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
hq : q = trinomial k m' n ↑x ↑v ↑z
h : p * p.mirror = q * q.mirror
hmul : ↑w * ↑u = ↑z * ↑x
hadd : eval 1 (trinomial k m n ↑u ↑v ↑w) ^ 2 = eval 1 (trinomial k m' n ↑x ↑v ↑z) ^ 2
⊢ q = p ∨ q = p.mirror

Tactic:
simp only [eval_add, eval_C_mul, eval_pow, eval_X, one_pow, mul_one, trinomial_def] at hadd

Result:
p q : ℤ[X]
k m m' n : ℕ
hkm : k < m
hmn : m < n
hkm' : k < m'
hmn' : m' < n
u v w x z : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
hq : q = trinomial k m' n ↑x ↑v ↑z
h : p * p.mirror = q * q.mirror
hmul : ↑w * ↑u = ↑z * ↑x
hadd : (↑u + ↑v + ↑w) ^ 2 = (↑x + ↑v + ↑z) ^ 2
⊢ q = p ∨ q = p.mirror

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

Tactic:
have hadd := congr_arg (eval 1) h

Result:
p q : ℤ[X]
k m m' n : ℕ
hkm : k < m
hmn : m < n
hkm' : k < m'
hmn' : m' < n
u v w x z : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
hq : q = trinomial k m' n ↑x ↑v ↑z
h : p * p.mirror = q * q.mirror
hmul : ↑w * ↑u = ↑z * ↑x
hadd : eval 1 (p * p.mirror) = eval 1 (q * q.mirror)
⊢ q = p ∨ q = p.mirror

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

Tactic:
exact
  irreducible_aux2 hkm hmn (lt_add_of_pos_left k (tsub_pos_of_lt hmn'))
    (lt_tsub_iff_right.mp ((tsub_lt_tsub_iff_left_of_le hmn'.le).mpr hkm')) u v w hp hq h

Result:
no goals

Proof state:
p q : ℤ[X]
k m m' n : ℕ
hkm : k < m
hmn : m < n
hkm' : k < m'
hmn' : m' < n
u v w x z : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
hq : q = trinomial k m' n ↑x ↑v ↑z
h : p * p.mirror = q * q.mirror
hmul : ↑w * ↑u = ↑z * ↑x
hadd : eval 1 (p * p.mirror) = eval 1 (q * q.mirror)
⊢ q = p ∨ q = p.mirror

Tactic:
rw [eval_mul, eval_mul, mirror_eval_one, mirror_eval_one, ← sq, ← sq, hp, hq] at hadd

Result:
p q : ℤ[X]
k m m' n : ℕ
hkm : k < m
hmn : m < n
hkm' : k < m'
hmn' : m' < n
u v w x z : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
hq : q = trinomial k m' n ↑x ↑v ↑z
h : p * p.mirror = q * q.mirror
hmul : ↑w * ↑u = ↑z * ↑x
hadd : eval 1 (trinomial k m n ↑u ↑v ↑w) ^ 2 = eval 1 (trinomial k m' n ↑x ↑v ↑z) ^ 2
⊢ q = p ∨ q = p.mirror

Proof state:
p q : ℤ[X]
k m m' n : ℕ
hkm : k < m
hmn : m < n
hkm' : k < m'
hmn' : m' < n
u v w x z : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
hq : q = trinomial k m' n ↑x ↑v ↑z
h : p * p.mirror = q * q.mirror
hmul : ↑w * ↑u = ↑z * ↑x
hadd : 2 * ↑w * ↑u + 2 * ↑v * (↑w + ↑u) = 2 * ↑z * ↑x + 2 * ↑v * (↑z + ↑x)
⊢ q = p ∨ q = p.mirror

Tactic:
rw [mul_assoc, hmul, ← mul_assoc, add_right_inj,
  mul_right_inj' (show 2 * (v : ℤ) ≠ 0 from mul_ne_zero two_ne_zero v.ne_zero)] at hadd

Result:
p q : ℤ[X]
k m m' n : ℕ
hkm : k < m
hmn : m < n
hkm' : k < m'
hmn' : m' < n
u v w x z : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
hq : q = trinomial k m' n ↑x ↑v ↑z
h : p * p.mirror = q * q.mirror
hmul : ↑w * ↑u = ↑z * ↑x
hadd : ↑w + ↑u = ↑z + ↑x
⊢ q = p ∨ q = p.mirror

Proof state:
p q : ℤ[X]
k m m' n : ℕ
hkm : k < m
hmn : m < n
hkm' : k < m'
hmn' : m' < n
u v w x z : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
hq : q = trinomial k m' n ↑x ↑v ↑z
h : p * p.mirror = q * q.mirror
hmul : ↑w * ↑u = ↑z * ↑x
hadd : (↑u + ↑v + ↑w) ^ 2 = (↑x + ↑v + ↑z) ^ 2
⊢ q = p ∨ q = p.mirror

Tactic:
rw [add_assoc, add_assoc, add_comm (u : ℤ), add_comm (x : ℤ), add_assoc, add_assoc] at hadd

Result:
p q : ℤ[X]
k m m' n : ℕ
hkm : k < m
hmn : m < n
hkm' : k < m'
hmn' : m' < n
u v w x z : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
hq : q = trinomial k m' n ↑x ↑v ↑z
h : p * p.mirror = q * q.mirror
hmul : ↑w * ↑u = ↑z * ↑x
hadd : (↑v + (↑w + ↑u)) ^ 2 = (↑v + (↑z + ↑x)) ^ 2
⊢ q = p ∨ q = p.mirror

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

Tactic:
have hmul := congr_arg leadingCoeff h

Result:
p q : ℤ[X]
k m m' n : ℕ
hkm : k < m
hmn : m < n
hkm' : k < m'
hmn' : m' < n
u v w x z : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
hq : q = trinomial k m' n ↑x ↑v ↑z
h : p * p.mirror = q * q.mirror
hmul : (p * p.mirror).leadingCoeff = (q * q.mirror).leadingCoeff
⊢ q = p ∨ q = p.mirror