awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (hypothesis)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
hq : q = trinomial k m' n (↑x : ℤ) (↑v : ℤ) (↑z : ℤ)
h : p * p.mirror = q * q.mirror
hmul : (↑w : ℤ) * (↑u : ℤ) = (↑z : ℤ) * (↑x : ℤ)
hadd : HPow.hPow (β := ℕ) ((↑v : ℤ) + ((↑w : ℤ) + (↑u : ℤ))) 2 = HPow.hPow (β := ℕ) ((↑v : ℤ) + ((↑z : ℤ) + (↑x : ℤ))) 2
⊢ q = p ∨ q = p.mirror
have hadd₁ : 2 * (↑w : ℤ) * (↑u : ℤ) + 2 * (↑v : ℤ) * ((↑w : ℤ) + (↑u : ℤ)) =
  2 * (↑z : ℤ) * (↑x : ℤ) + 2 * (↑v : ℤ) * ((↑z : ℤ) + (↑x : ℤ))
Tactic: simp only [add_sq', add_assoc, add_right_inj, ← Units.val_pow_eq_pow_val, Int.units_sq] at hadd
Cleared hypotheses: hadd
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (hypothesis)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
hq : q = trinomial k m' n (↑x : ℤ) (↑v : ℤ) (↑z : ℤ)
h : p * p.mirror = q * q.mirror
hmul : (↑w : ℤ) * (↑u : ℤ) = (↑z : ℤ) * (↑x : ℤ)
hadd :
  2 * (↑w : ℤ) * (↑u : ℤ) + 2 * (↑v : ℤ) * ((↑w : ℤ) + (↑u : ℤ)) =
    2 * (↑z : ℤ) * (↑x : ℤ) + 2 * (↑v : ℤ) * ((↑z : ℤ) + (↑x : ℤ))
⊢ q = p ∨ q = p.mirror
have hadd₁ : (↑w : ℤ) + (↑u : ℤ) = (↑z : ℤ) + (↑x : ℤ)
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
Cleared hypotheses: hadd
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (hypothesis)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
hq : q = trinomial k m' n (↑x : ℤ) (↑v : ℤ) (↑z : ℤ)
h : p * p.mirror = q * q.mirror
hmul : (↑w : ℤ) * (↑u : ℤ) = (↑z : ℤ) * (↑x : ℤ)
hadd : (↑w : ℤ) + (↑u : ℤ) = (↑z : ℤ) + (↑x : ℤ)
⊢ q = p ∨ q = p.mirror
have hadd₁ : (↑w : ℤ) = (↑z : ℤ) ∧ (↑u : ℤ) = (↑x : ℤ) ∨ (↑w : ℤ) = (↑x : ℤ) ∧ (↑u : ℤ) = (↑z : ℤ)
Tactic: replace hadd :=
    (Int.isUnit_add_isUnit_eq_isUnit_add_isUnit w.isUnit u.isUnit z.isUnit x.isUnit).mp hadd
Cleared hypotheses: hadd
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (hypothesis)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
hq : q = trinomial k m' n (↑x : ℤ) (↑v : ℤ) (↑z : ℤ)
h : p * p.mirror = q * q.mirror
hmul : (↑w : ℤ) * (↑u : ℤ) = (↑z : ℤ) * (↑x : ℤ)
hadd : (↑w : ℤ) = (↑z : ℤ) ∧ (↑u : ℤ) = (↑x : ℤ) ∨ (↑w : ℤ) = (↑x : ℤ) ∧ (↑u : ℤ) = (↑z : ℤ)
⊢ q = p ∨ q = p.mirror
have hadd₁ : w = z ∧ u = x ∨ w = x ∧ u = z
Tactic: simp only [Units.val_inj] at hadd
Cleared hypotheses: hadd
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
hq : q = trinomial k m' n (↑x : ℤ) (↑v : ℤ) (↑z : ℤ)
h : p * p.mirror = q * q.mirror
hmul : (↑w : ℤ) * (↑u : ℤ) = (↑z : ℤ) * (↑x : ℤ)
hadd : w = z ∧ u = x ∨ w = x ∧ u = z
⊢ q = p ∨ q = p.mirror
have inl : ∀ (hq : q = trinomial k m' n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) (hmul : (↑w : ℤ) * (↑u : ℤ) = (↑w : ℤ) * (↑u : ℤ)),
  q = p ∨ q = p.mirror
Tactic: rcases hadd with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
Cleared hypotheses: x, z, hq, hmul, hadd
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
hq : q = trinomial k m' n (↑x : ℤ) (↑v : ℤ) (↑z : ℤ)
h : p * p.mirror = q * q.mirror
hmul : (↑w : ℤ) * (↑u : ℤ) = (↑z : ℤ) * (↑x : ℤ)
hadd : w = z ∧ u = x ∨ w = x ∧ u = z
inl :
  ∀ (hq : q = trinomial k m' n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) (hmul : (↑w : ℤ) * (↑u : ℤ) = (↑w : ℤ) * (↑u : ℤ)),
    q = p ∨ q = p.mirror
⊢ q = p ∨ q = p.mirror
have inr : ∀ (hq : q = trinomial k m' n (↑w : ℤ) (↑v : ℤ) (↑u : ℤ)) (hmul : (↑w : ℤ) * (↑u : ℤ) = (↑u : ℤ) * (↑w : ℤ)),
  q = p ∨ q = p.mirror
Tactic: rcases hadd with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
Cleared hypotheses: x, z, hq, hmul, hadd
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (hypothesis)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
h : p * p.mirror = q * q.mirror
hq : q = trinomial k m' n (↑w : ℤ) (↑v : ℤ) (↑u : ℤ)
hmul : (↑w : ℤ) * (↑u : ℤ) = (↑u : ℤ) * (↑w : ℤ)
⊢ q = p ∨ q = p.mirror
have hq₁ : q.mirror = trinomial k (n - m' + k) n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)
Tactic: rw [← mirror_inj, trinomial_mirror hkm' hmn' w.ne_zero u.ne_zero] at hq
Cleared hypotheses: hq, hmul
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (hypothesis)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
h : p * p.mirror = q * q.mirror
hq : q.mirror = trinomial k (n - m' + k) n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)
hmul : (↑w : ℤ) * (↑u : ℤ) = (↑u : ℤ) * (↑w : ℤ)
⊢ q = p ∨ q = p.mirror
have h₁ : p * p.mirror = q.mirror * q.mirror.mirror
Tactic: rw [mul_comm q, ← q.mirror_mirror, q.mirror.mirror_mirror] at h
Cleared hypotheses: h, hq, hmul
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
h : p * p.mirror = q.mirror * q.mirror.mirror
hq : q.mirror = trinomial k (n - m' + k) n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)
hmul : (↑w : ℤ) * (↑u : ℤ) = (↑u : ℤ) * (↑w : ℤ)
⊢ q = p ∨ q = p.mirror
have inr : q.mirror = p ∨ q.mirror = p.mirror
Tactic: rw [← mirror_inj, or_comm, ← mirror_eq_iff]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
p : ℤ[X]
hp : p.IsUnitTrinomial
h : IsRelPrime p p.mirror
⊢ Irreducible p
have refine : ∀ (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
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (hypothesis)
Goal:
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
have hq : q.IsUnitTrinomial
Tactic: have hq : IsUnitTrinomial q := (isUnitTrinomial_iff'' hpq).mp hp
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
p : 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
have obtain : ∀ (k m n : ℕ) (hkm : k < m) (hmn : m < n) (u v w : ℤˣ) (hp : p = trinomial k m n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)),
  q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror
Tactic: obtain ⟨k, m, n, hkm, hmn, u, v, w, hp⟩ := hp
Cleared hypotheses: hp
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror
have obtain : ∀ (k' m' n' : ℕ) (hkm' : k' < m') (hmn' : m' < n') (x y z : ℤˣ)
  (hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)), q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror
Tactic: obtain ⟨k', m', n', hkm', hmn', x, y, z, hq⟩ := hq
Cleared hypotheses: hq
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
⊢ k = k'
have rw : 2 * k = 2 * k'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
rw : 2 * k = 2 * k'
⊢ k = k'
have rw₁ : 2 * natTrailingDegree (trinomial k m n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) = 2 * k'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
rw : 2 * k = 2 * k'
rw₁ : 2 * natTrailingDegree (trinomial k m n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) = 2 * k'
⊢ k = k'
have rw₂ : ℤ
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
rw : 2 * k = 2 * k'
rw₁ : 2 * natTrailingDegree (trinomial k m n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) = 2 * k'
rw₂ rw₃ : ℤ
⊢ k = k'
have rw₄ : 2 * p.natTrailingDegree = 2 * k'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
rw : 2 * k = 2 * k'
rw₁ : 2 * natTrailingDegree (trinomial k m n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) = 2 * k'
rw₂ rw₃ : ℤ
rw₄ : 2 * p.natTrailingDegree = 2 * k'
⊢ k = k'
have rw₅ : (p * p.mirror).natTrailingDegree = 2 * k'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
rw : 2 * k = 2 * k'
rw₁ : 2 * natTrailingDegree (trinomial k m n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) = 2 * k'
rw₂ rw₃ : ℤ
rw₄ : 2 * p.natTrailingDegree = 2 * k'
rw₅ : (p * p.mirror).natTrailingDegree = 2 * k'
⊢ k = k'
have rw₆ : (q * q.mirror).natTrailingDegree = 2 * k'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
rw : 2 * k = 2 * k'
rw₁ : 2 * natTrailingDegree (trinomial k m n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) = 2 * k'
rw₂ rw₃ : ℤ
rw₄ : 2 * p.natTrailingDegree = 2 * k'
rw₅ : (p * p.mirror).natTrailingDegree = 2 * k'
rw₆ : (q * q.mirror).natTrailingDegree = 2 * k'
⊢ k = k'
have rw₇ : 2 * q.natTrailingDegree = 2 * k'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
rw : 2 * k = 2 * k'
rw₁ : 2 * natTrailingDegree (trinomial k m n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) = 2 * k'
rw₂ rw₃ : ℤ
rw₄ : 2 * p.natTrailingDegree = 2 * k'
rw₅ : (p * p.mirror).natTrailingDegree = 2 * k'
rw₆ : (q * q.mirror).natTrailingDegree = 2 * k'
rw₇ : 2 * q.natTrailingDegree = 2 * k'
⊢ k = k'
have rw₈ : 2 * natTrailingDegree (trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)) = 2 * k'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
rw : 2 * k = 2 * k'
rw₁ : 2 * natTrailingDegree (trinomial k m n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) = 2 * k'
rw₂ rw₃ : ℤ
rw₄ : 2 * p.natTrailingDegree = 2 * k'
rw₅ : (p * p.mirror).natTrailingDegree = 2 * k'
rw₆ : (q * q.mirror).natTrailingDegree = 2 * k'
rw₇ : 2 * q.natTrailingDegree = 2 * k'
rw₈ : 2 * natTrailingDegree (trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)) = 2 * k'
⊢ k = k'
have rw₉ : 2 * k' = 2 * k'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (hypothesis)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror
have hk : k = k'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
hk : k = k'
⊢ n = n'
have rw : 2 * n = 2 * n'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
hk : k = k'
rw : 2 * n = 2 * n'
⊢ n = n'
have rw₁ : 2 * natDegree (trinomial k m n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) = 2 * n'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
hk : k = k'
rw : 2 * n = 2 * n'
rw₁ : 2 * natDegree (trinomial k m n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) = 2 * n'
⊢ n = n'
have rw₂ : ℤ
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
hk : k = k'
rw : 2 * n = 2 * n'
rw₁ : 2 * natDegree (trinomial k m n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) = 2 * n'
rw₂ rw₃ : ℤ
⊢ n = n'
have rw₄ : 2 * p.natDegree = 2 * n'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
hk : k = k'
rw : 2 * n = 2 * n'
rw₁ : 2 * natDegree (trinomial k m n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) = 2 * n'
rw₂ rw₃ : ℤ
rw₄ : 2 * p.natDegree = 2 * n'
⊢ n = n'
have rw₅ : (p * p.mirror).natDegree = 2 * n'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
hk : k = k'
rw : 2 * n = 2 * n'
rw₁ : 2 * natDegree (trinomial k m n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) = 2 * n'
rw₂ rw₃ : ℤ
rw₄ : 2 * p.natDegree = 2 * n'
rw₅ : (p * p.mirror).natDegree = 2 * n'
⊢ n = n'
have rw₆ : (q * q.mirror).natDegree = 2 * n'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
hk : k = k'
rw : 2 * n = 2 * n'
rw₁ : 2 * natDegree (trinomial k m n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) = 2 * n'
rw₂ rw₃ : ℤ
rw₄ : 2 * p.natDegree = 2 * n'
rw₅ : (p * p.mirror).natDegree = 2 * n'
rw₆ : (q * q.mirror).natDegree = 2 * n'
⊢ n = n'
have rw₇ : 2 * q.natDegree = 2 * n'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
hk : k = k'
rw : 2 * n = 2 * n'
rw₁ : 2 * natDegree (trinomial k m n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) = 2 * n'
rw₂ rw₃ : ℤ
rw₄ : 2 * p.natDegree = 2 * n'
rw₅ : (p * p.mirror).natDegree = 2 * n'
rw₆ : (q * q.mirror).natDegree = 2 * n'
rw₇ : 2 * q.natDegree = 2 * n'
⊢ n = n'
have rw₈ : 2 * natDegree (trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)) = 2 * n'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (goal)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
hk : k = k'
rw : 2 * n = 2 * n'
rw₁ : 2 * natDegree (trinomial k m n (↑u : ℤ) (↑v : ℤ) (↑w : ℤ)) = 2 * n'
rw₂ rw₃ : ℤ
rw₄ : 2 * p.natDegree = 2 * n'
rw₅ : (p * p.mirror).natDegree = 2 * n'
rw₆ : (q * q.mirror).natDegree = 2 * n'
rw₇ : 2 * q.natDegree = 2 * n'
rw₈ : 2 * natDegree (trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)) = 2 * n'
⊢ n = n'
have rw₉ : 2 * n' = 2 * n'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (hypothesis)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
hk : k = k'
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror
have hn : n = n'
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.UnitTrinomial (hypothesis)
Goal:
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 : ℤ) (↑v : ℤ) (↑w : ℤ)
k' m' n' : ℕ
hkm' : k' < m'
hmn' : m' < n'
x y z : ℤˣ
hq : q = trinomial k' m' n' (↑x : ℤ) (↑y : ℤ) (↑z : ℤ)
hk : k = k'
hn : n = n'
⊢ q = p ∨ q = -p ∨ q = p.mirror ∨ q = -p.mirror
have hkm'₁ : k < m'
Tactic: subst hk
Cleared hypotheses: k', hkm', hq, hk