l3lab/ntp-mathlib huggingface dataset:
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 := by
  have hmul := congr_arg leadingCoeff h
  rw [leadingCoeff_mul, leadingCoeff_mul, mirror_leadingCoeff, mirror_leadingCoeff, hp, hq,
    trinomial_leadingCoeff hkm hmn w.ne_zero, trinomial_leadingCoeff hkm' hmn' z.ne_zero,
    trinomial_trailingCoeff hkm hmn u.ne_zero, trinomial_trailingCoeff hkm' hmn' x.ne_zero]
    at hmul
  have hadd := congr_arg (eval 1) h
  rw [eval_mul, eval_mul, mirror_eval_one, mirror_eval_one, ← sq, ← sq, hp, hq] at hadd
  simp only [eval_add, eval_C_mul, eval_pow, eval_X, one_pow, mul_one, trinomial_def] at hadd
  rw [add_assoc, add_assoc, add_comm (u : ℤ), add_comm (x : ℤ), add_assoc, add_assoc] at hadd
  simp only [add_sq', add_assoc, add_right_inj, ← Units.val_pow_eq_pow_val, Int.units_sq] at hadd
  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
  replace hadd :=
    (Int.isUnit_add_isUnit_eq_isUnit_add_isUnit w.isUnit u.isUnit z.isUnit x.isUnit).mp hadd
  simp only [Units.eq_iff] at hadd
  

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 * mirror p = q * mirror q
hmul : ↑w * ↑u = ↑z * ↑x
hadd : w = z ∧ u = x ∨ w = x ∧ u = z
⊢ q = p ∨ q = mirror p

Next tactic:
rcases hadd with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
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 := by
  have hmul := congr_arg leadingCoeff h
  rw [leadingCoeff_mul, leadingCoeff_mul, mirror_leadingCoeff, mirror_leadingCoeff, hp, hq,
    trinomial_leadingCoeff hkm hmn w.ne_zero, trinomial_leadingCoeff hkm' hmn' z.ne_zero,
    trinomial_trailingCoeff hkm hmn u.ne_zero, trinomial_trailingCoeff hkm' hmn' x.ne_zero]
    at hmul
  have hadd := congr_arg (eval 1) h
  rw [eval_mul, eval_mul, mirror_eval_one, mirror_eval_one, ← sq, ← sq, hp, hq] at hadd
  simp only [eval_add, eval_C_mul, eval_pow, eval_X, one_pow, mul_one, trinomial_def] at hadd
  rw [add_assoc, add_assoc, add_comm (u : ℤ), add_comm (x : ℤ), add_assoc, add_assoc] at hadd
  simp only [add_sq', add_assoc, add_right_inj, ← Units.val_pow_eq_pow_val, Int.units_sq] at hadd
  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
  replace hadd :=
    (Int.isUnit_add_isUnit_eq_isUnit_add_isUnit w.isUnit u.isUnit z.isUnit x.isUnit).mp hadd
  simp only [Units.eq_iff] at hadd
  rcases hadd with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
  · 

Proof state:
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 * mirror p = q * mirror q
hq : q = trinomial k m' n ↑u ↑v ↑w
hmul : ↑w * ↑u = ↑w * ↑u
⊢ q = p ∨ q = mirror p

Next tactic:
exact irreducible_aux2 hkm hmn hkm' hmn' u v w hp hq h
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 := by
  have hmul := congr_arg leadingCoeff h
  rw [leadingCoeff_mul, leadingCoeff_mul, mirror_leadingCoeff, mirror_leadingCoeff, hp, hq,
    trinomial_leadingCoeff hkm hmn w.ne_zero, trinomial_leadingCoeff hkm' hmn' z.ne_zero,
    trinomial_trailingCoeff hkm hmn u.ne_zero, trinomial_trailingCoeff hkm' hmn' x.ne_zero]
    at hmul
  have hadd := congr_arg (eval 1) h
  rw [eval_mul, eval_mul, mirror_eval_one, mirror_eval_one, ← sq, ← sq, hp, hq] at hadd
  simp only [eval_add, eval_C_mul, eval_pow, eval_X, one_pow, mul_one, trinomial_def] at hadd
  rw [add_assoc, add_assoc, add_comm (u : ℤ), add_comm (x : ℤ), add_assoc, add_assoc] at hadd
  simp only [add_sq', add_assoc, add_right_inj, ← Units.val_pow_eq_pow_val, Int.units_sq] at hadd
  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
  replace hadd :=
    (Int.isUnit_add_isUnit_eq_isUnit_add_isUnit w.isUnit u.isUnit z.isUnit x.isUnit).mp hadd
  simp only [Units.eq_iff] at hadd
  rcases hadd with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
  · exact irreducible_aux2 hkm hmn hkm' hmn' u v w hp hq h
  · 

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 * mirror p = q * mirror q
hq : q = trinomial k m' n ↑w ↑v ↑u
hmul : ↑w * ↑u = ↑u * ↑w
⊢ q = p ∨ q = mirror p

Next tactic:
rw [← mirror_inj, trinomial_mirror hkm' hmn' w.ne_zero u.ne_zero] at hq
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 := by
  have hmul := congr_arg leadingCoeff h
  rw [leadingCoeff_mul, leadingCoeff_mul, mirror_leadingCoeff, mirror_leadingCoeff, hp, hq,
    trinomial_leadingCoeff hkm hmn w.ne_zero, trinomial_leadingCoeff hkm' hmn' z.ne_zero,
    trinomial_trailingCoeff hkm hmn u.ne_zero, trinomial_trailingCoeff hkm' hmn' x.ne_zero]
    at hmul
  have hadd := congr_arg (eval 1) h
  rw [eval_mul, eval_mul, mirror_eval_one, mirror_eval_one, ← sq, ← sq, hp, hq] at hadd
  simp only [eval_add, eval_C_mul, eval_pow, eval_X, one_pow, mul_one, trinomial_def] at hadd
  rw [add_assoc, add_assoc, add_comm (u : ℤ), add_comm (x : ℤ), add_assoc, add_assoc] at hadd
  simp only [add_sq', add_assoc, add_right_inj, ← Units.val_pow_eq_pow_val, Int.units_sq] at hadd
  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
  replace hadd :=
    (Int.isUnit_add_isUnit_eq_isUnit_add_isUnit w.isUnit u.isUnit z.isUnit x.isUnit).mp hadd
  simp only [Units.eq_iff] at hadd
  rcases hadd with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
  · exact irreducible_aux2 hkm hmn hkm' hmn' u v w hp hq h
  · rw [← mirror_inj, trinomial_mirror hkm' hmn' w.ne_zero u.ne_zero] at hq
    

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 * mirror p = q * mirror q
hq : mirror q = trinomial k (n - m' + k) n ↑u ↑v ↑w
hmul : ↑w * ↑u = ↑u * ↑w
⊢ q = p ∨ q = mirror p

Next tactic:
rw [mul_comm q, ← q.mirror_mirror, q.mirror.mirror_mirror] at h
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 := by
  have hmul := congr_arg leadingCoeff h
  rw [leadingCoeff_mul, leadingCoeff_mul, mirror_leadingCoeff, mirror_leadingCoeff, hp, hq,
    trinomial_leadingCoeff hkm hmn w.ne_zero, trinomial_leadingCoeff hkm' hmn' z.ne_zero,
    trinomial_trailingCoeff hkm hmn u.ne_zero, trinomial_trailingCoeff hkm' hmn' x.ne_zero]
    at hmul
  have hadd := congr_arg (eval 1) h
  rw [eval_mul, eval_mul, mirror_eval_one, mirror_eval_one, ← sq, ← sq, hp, hq] at hadd
  simp only [eval_add, eval_C_mul, eval_pow, eval_X, one_pow, mul_one, trinomial_def] at hadd
  rw [add_assoc, add_assoc, add_comm (u : ℤ), add_comm (x : ℤ), add_assoc, add_assoc] at hadd
  simp only [add_sq', add_assoc, add_right_inj, ← Units.val_pow_eq_pow_val, Int.units_sq] at hadd
  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
  replace hadd :=
    (Int.isUnit_add_isUnit_eq_isUnit_add_isUnit w.isUnit u.isUnit z.isUnit x.isUnit).mp hadd
  simp only [Units.eq_iff] at hadd
  rcases hadd with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
  · exact irreducible_aux2 hkm hmn hkm' hmn' u v w hp hq h
  · rw [← mirror_inj, trinomial_mirror hkm' hmn' w.ne_zero u.ne_zero] at hq
    rw [mul_comm q, ← q.mirror_mirror, q.mirror.mirror_mirror] at h
    

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 * mirror p = mirror q * mirror (mirror q)
hq : mirror q = trinomial k (n - m' + k) n ↑u ↑v ↑w
hmul : ↑w * ↑u = ↑u * ↑w
⊢ q = p ∨ q = mirror p

Next tactic:
rw [← mirror_inj, or_comm, ← mirror_eq_iff]
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 := by
  have hmul := congr_arg leadingCoeff h
  rw [leadingCoeff_mul, leadingCoeff_mul, mirror_leadingCoeff, mirror_leadingCoeff, hp, hq,
    trinomial_leadingCoeff hkm hmn w.ne_zero, trinomial_leadingCoeff hkm' hmn' z.ne_zero,
    trinomial_trailingCoeff hkm hmn u.ne_zero, trinomial_trailingCoeff hkm' hmn' x.ne_zero]
    at hmul
  have hadd := congr_arg (eval 1) h
  rw [eval_mul, eval_mul, mirror_eval_one, mirror_eval_one, ← sq, ← sq, hp, hq] at hadd
  simp only [eval_add, eval_C_mul, eval_pow, eval_X, one_pow, mul_one, trinomial_def] at hadd
  rw [add_assoc, add_assoc, add_comm (u : ℤ), add_comm (x : ℤ), add_assoc, add_assoc] at hadd
  simp only [add_sq', add_assoc, add_right_inj, ← Units.val_pow_eq_pow_val, Int.units_sq] at hadd
  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
  replace hadd :=
    (Int.isUnit_add_isUnit_eq_isUnit_add_isUnit w.isUnit u.isUnit z.isUnit x.isUnit).mp hadd
  simp only [Units.eq_iff] at hadd
  rcases hadd with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
  · exact irreducible_aux2 hkm hmn hkm' hmn' u v w hp hq h
  · rw [← mirror_inj, trinomial_mirror hkm' hmn' w.ne_zero u.ne_zero] at hq
    rw [mul_comm q, ← q.mirror_mirror, q.mirror.mirror_mirror] at h
    rw [← mirror_inj, or_comm, ← mirror_eq_iff]
    

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 * mirror p = mirror q * mirror (mirror q)
hq : mirror q = trinomial k (n - m' + k) n ↑u ↑v ↑w
hmul : ↑w * ↑u = ↑u * ↑w
⊢ mirror q = p ∨ mirror q = mirror p

Next 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
theorem irreducible_of_coprime (hp : p.IsUnitTrinomial)
    (h : ∀ q : ℤ[X], q ∣ p → q ∣ p.mirror → IsUnit q) : Irreducible p := by
  

Proof state:
p q : ℤ[X]
hp : IsUnitTrinomial p
h : ∀ (q : ℤ[X]), q ∣ p → q ∣ mirror p → IsUnit q
⊢ Irreducible p

Next tactic:
refine' irreducible_of_mirror hp.not_isUnit (fun q hpq => _) h
theorem irreducible_of_coprime (hp : p.IsUnitTrinomial)
    (h : ∀ q : ℤ[X], q ∣ p → q ∣ p.mirror → IsUnit q) : Irreducible p := by
  refine' irreducible_of_mirror hp.not_isUnit (fun q hpq => _) h
  

Proof state:
p q✝ : ℤ[X]
hp : IsUnitTrinomial p
h : ∀ (q : ℤ[X]), q ∣ p → q ∣ mirror p → IsUnit q
q : ℤ[X]
hpq : p * mirror p = q * mirror q
⊢ q = p ∨ q = -p ∨ q = mirror p ∨ q = -mirror p

Next tactic:
have hq : IsUnitTrinomial q := (isUnitTrinomial_iff'' hpq).mp hp
theorem irreducible_of_coprime (hp : p.IsUnitTrinomial)
    (h : ∀ q : ℤ[X], q ∣ p → q ∣ p.mirror → IsUnit q) : Irreducible p := by
  refine' irreducible_of_mirror hp.not_isUnit (fun q hpq => _) h
  have hq : IsUnitTrinomial q := (isUnitTrinomial_iff'' hpq).mp hp
  

Proof state:
p q✝ : ℤ[X]
hp : IsUnitTrinomial p
h : ∀ (q : ℤ[X]), q ∣ p → q ∣ mirror p → IsUnit q
q : ℤ[X]
hpq : p * mirror p = q * mirror q
hq : IsUnitTrinomial q
⊢ q = p ∨ q = -p ∨ q = mirror p ∨ q = -mirror p

Next tactic:
obtain ⟨k, m, n, hkm, hmn, u, v, w, hp⟩ := hp
theorem irreducible_of_coprime (hp : p.IsUnitTrinomial)
    (h : ∀ q : ℤ[X], q ∣ p → q ∣ p.mirror → IsUnit q) : Irreducible p := by
  refine' irreducible_of_mirror hp.not_isUnit (fun q hpq => _) h
  have hq : IsUnitTrinomial q := (isUnitTrinomial_iff'' hpq).mp hp
  obtain ⟨k, m, n, hkm, hmn, u, v, w, hp⟩ := hp
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : ℤ[X]
h : ∀ (q : ℤ[X]), q ∣ p → q ∣ mirror p → IsUnit q
q : ℤ[X]
hpq : p * mirror p = q * mirror q
hq : IsUnitTrinomial q
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 = mirror p ∨ q = -mirror p

Next tactic:
obtain ⟨k', m', n', hkm', hmn', x, y, z, hq⟩ := hq
theorem irreducible_of_coprime (hp : p.IsUnitTrinomial)
    (h : ∀ q : ℤ[X], q ∣ p → q ∣ p.mirror → IsUnit q) : Irreducible p := by
  refine' irreducible_of_mirror hp.not_isUnit (fun q hpq => _) h
  have hq : IsUnitTrinomial q := (isUnitTrinomial_iff'' hpq).mp hp
  obtain ⟨k, m, n, hkm, hmn, u, v, w, hp⟩ := hp
  obtain ⟨k', m', n', hkm', hmn', x, y, z, hq⟩ := hq
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : ℤ[X]
h : ∀ (q : ℤ[X]), q ∣ p → q ∣ mirror p → IsUnit q
q : ℤ[X]
hpq : p * mirror p = q * mirror q
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 = mirror p ∨ q = -mirror p

Next 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]
theorem irreducible_of_coprime (hp : p.IsUnitTrinomial)
    (h : ∀ q : ℤ[X], q ∣ p → q ∣ p.mirror → IsUnit q) : Irreducible p := by
  refine' irreducible_of_mirror hp.not_isUnit (fun q hpq => _) h
  have hq : IsUnitTrinomial q := (isUnitTrinomial_iff'' hpq).mp hp
  obtain ⟨k, m, n, hkm, hmn, u, v, w, hp⟩ := hp
  obtain ⟨k', m', n', hkm', hmn', x, y, z, hq⟩ := hq
  have hk : k = k' := by
    

Proof state:
p q✝ : ℤ[X]
h : ∀ (q : ℤ[X]), q ∣ p → q ∣ mirror p → IsUnit q
q : ℤ[X]
hpq : p * mirror p = q * mirror q
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
⊢ k = k'

Next tactic:
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]
theorem irreducible_of_coprime (hp : p.IsUnitTrinomial)
    (h : ∀ q : ℤ[X], q ∣ p → q ∣ p.mirror → IsUnit q) : Irreducible p := by
  refine' irreducible_of_mirror hp.not_isUnit (fun q hpq => _) h
  have hq : IsUnitTrinomial q := (isUnitTrinomial_iff'' hpq).mp hp
  obtain ⟨k, m, n, hkm, hmn, u, v, w, hp⟩ := hp
  obtain ⟨k', m', n', hkm', hmn', x, y, z, hq⟩ := hq
  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]
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : ℤ[X]
h : ∀ (q : ℤ[X]), q ∣ p → q ∣ mirror p → IsUnit q
q : ℤ[X]
hpq : p * mirror p = q * mirror q
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 = mirror p ∨ q = -mirror p

Next 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]
theorem irreducible_of_coprime (hp : p.IsUnitTrinomial)
    (h : ∀ q : ℤ[X], q ∣ p → q ∣ p.mirror → IsUnit q) : Irreducible p := by
  refine' irreducible_of_mirror hp.not_isUnit (fun q hpq => _) h
  have hq : IsUnitTrinomial q := (isUnitTrinomial_iff'' hpq).mp hp
  obtain ⟨k, m, n, hkm, hmn, u, v, w, hp⟩ := hp
  obtain ⟨k', m', n', hkm', hmn', x, y, z, hq⟩ := hq
  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]
  have hn : n = n' := by
    

Proof state:
p q✝ : ℤ[X]
h : ∀ (q : ℤ[X]), q ∣ p → q ∣ mirror p → IsUnit q
q : ℤ[X]
hpq : p * mirror p = q * mirror q
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'
⊢ n = n'

Next tactic:
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]
theorem irreducible_of_coprime (hp : p.IsUnitTrinomial)
    (h : ∀ q : ℤ[X], q ∣ p → q ∣ p.mirror → IsUnit q) : Irreducible p := by
  refine' irreducible_of_mirror hp.not_isUnit (fun q hpq => _) h
  have hq : IsUnitTrinomial q := (isUnitTrinomial_iff'' hpq).mp hp
  obtain ⟨k, m, n, hkm, hmn, u, v, w, hp⟩ := hp
  obtain ⟨k', m', n', hkm', hmn', x, y, z, hq⟩ := hq
  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]
  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]
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : ℤ[X]
h : ∀ (q : ℤ[X]), q ∣ p → q ∣ mirror p → IsUnit q
q : ℤ[X]
hpq : p * mirror p = q * mirror q
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 = mirror p ∨ q = -mirror p

Next tactic:
subst hk
theorem irreducible_of_coprime (hp : p.IsUnitTrinomial)
    (h : ∀ q : ℤ[X], q ∣ p → q ∣ p.mirror → IsUnit q) : Irreducible p := by
  refine' irreducible_of_mirror hp.not_isUnit (fun q hpq => _) h
  have hq : IsUnitTrinomial q := (isUnitTrinomial_iff'' hpq).mp hp
  obtain ⟨k, m, n, hkm, hmn, u, v, w, hp⟩ := hp
  obtain ⟨k', m', n', hkm', hmn', x, y, z, hq⟩ := hq
  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]
  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]
  subst hk
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : ℤ[X]
h : ∀ (q : ℤ[X]), q ∣ p → q ∣ mirror p → IsUnit q
q : ℤ[X]
hpq : p * mirror p = q * mirror q
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 = mirror p ∨ q = -mirror p

Next tactic:
subst hn
theorem irreducible_of_coprime (hp : p.IsUnitTrinomial)
    (h : ∀ q : ℤ[X], q ∣ p → q ∣ p.mirror → IsUnit q) : Irreducible p := by
  refine' irreducible_of_mirror hp.not_isUnit (fun q hpq => _) h
  have hq : IsUnitTrinomial q := (isUnitTrinomial_iff'' hpq).mp hp
  obtain ⟨k, m, n, hkm, hmn, u, v, w, hp⟩ := hp
  obtain ⟨k', m', n', hkm', hmn', x, y, z, hq⟩ := hq
  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]
  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]
  subst hk
  subst hn
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
p q✝ : ℤ[X]
h : ∀ (q : ℤ[X]), q ∣ p → q ∣ mirror p → IsUnit q
q : ℤ[X]
hpq : p * mirror p = q * mirror q
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 = mirror p ∨ q = -mirror p

Next 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)
theorem irreducible_of_coprime (hp : p.IsUnitTrinomial)
    (h : ∀ q : ℤ[X], q ∣ p → q ∣ p.mirror → IsUnit q) : Irreducible p := by
  refine' irreducible_of_mirror hp.not_isUnit (fun q hpq => _) h
  have hq : IsUnitTrinomial q := (isUnitTrinomial_iff'' hpq).mp hp
  obtain ⟨k, m, n, hkm, hmn, u, v, w, hp⟩ := hp
  obtain ⟨k', m', n', hkm', hmn', x, y, z, hq⟩ := hq
  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]
  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]
  subst hk
  subst hn
  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)
  · -- Porting note: `rw [h1] at *` rewrites at `h1`
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inl
p q✝ : ℤ[X]
h : ∀ (q : ℤ[X]), q ∣ p → q ∣ mirror p → IsUnit q
q : ℤ[X]
hpq : p * mirror p = q * mirror q
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 = mirror p ∨ q = -mirror p

Next tactic:
rw [h1] at hq