awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
this : Eq (α := ℂ) (∑ x ∈ s, weight x • (z - x)) 0
z₁ : z = s.centerMass weight fun (x : ℂ) ↦ z
z₂ : (s.centerMass weight fun (x : ℂ) ↦ z) = (s.centerMass weight fun (x : ℂ) ↦ z - x) + s.centerMass weight id
⊢ z = s.centerMass weight id
have z₃ : (s.centerMass weight fun (x : ℂ) ↦ z - x) + s.centerMass weight id = s.centerMass weight id
Tactic: calc
    z = s.centerMass weight fun _ ↦ z := by
      rw [Finset.centerMass, ← Finset.sum_smul, inv_smul_smul₀]
      exact (sum_derivRootWeight_pos hP z).ne'
    _ = s.centerMass weight (z - ·) + s.centerMass weight id := by
      simp only [Finset.centerMass, ← smul_add, ← Finset.sum_add_distrib, id, sub_add_cancel]
    _ = s.centerMass weight id := by
      simp only [add_eq_right, Finset.centerMass, this, smul_zero]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
⊢ z = s.centerMass weight id
have suffices : Eq (α := ℂ) (∑ x ∈ s, weight x • (z - x)) 0
Tactic: suffices ∑ x ∈ s, weight x • (z - x) = 0 by calc
    z = s.centerMass weight fun _ ↦ z := by
      rw [Finset.centerMass, ← Finset.sum_smul, inv_smul_smul₀]
      exact (sum_derivRootWeight_pos hP z).ne'
    _ = s.centerMass weight (z - ·) + s.centerMass weight id := by
      simp only [Finset.centerMass, ← smul_add, ← Finset.sum_add_distrib, id, sub_add_cancel]
    _ = s.centerMass weight id := by
      simp only [add_eq_right, Finset.centerMass, this, smul_zero]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
⊢ Eq (α := ℂ) (∑ x ∈ s, weight x • (z - x)) 0
have pos : ∀ (hzP : eval z P = 0), Eq (α := ℂ) (∑ x ∈ s, weight x • (z - x)) 0
Tactic: by_cases hzP : P.eval z = 0
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
pos : ∀ (hzP : eval z P = 0), Eq (α := ℂ) (∑ x ∈ s, weight x • (z - x)) 0
⊢ Eq (α := ℂ) (∑ x ∈ s, weight x • (z - x)) 0
have neg : ∀ (hzP : ¬eval z P = 0), Eq (α := ℂ) (∑ x ∈ s, weight x • (z - x)) 0
Tactic: by_cases hzP : P.eval z = 0
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : eval z P = 0
⊢ Eq (α := ℂ) (∑ x ∈ s, weight x • (z - x)) 0
have pos : Eq (α := ℂ) (∑ x ∈ s, HSMul.hSMul (α := ℝ) (Pi.single (M := fun (a : ℂ) ↦ ℝ) z 1 x : ℝ) (z - x)) 0
Tactic: simp only [weight, derivRootWeight, if_pos hzP]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : eval z P = 0
⊢ Eq (α := ℂ) (∑ x ∈ s, HSMul.hSMul (α := ℝ) (Pi.single (M := fun (a : ℂ) ↦ ℝ) z 1 x : ℝ) (z - x)) 0
have pos : HSMul.hSMul (α := ℝ) (Pi.single (M := fun (a : ℂ) ↦ ℝ) z 1 z : ℝ) (z - z) = 0
Tactic: rw [Finset.sum_eq_single z]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : eval z P = 0
pos : HSMul.hSMul (α := ℝ) (Pi.single (M := fun (a : ℂ) ↦ ℝ) z 1 z : ℝ) (z - z) = 0
⊢ Eq (α := ℂ) (∑ x ∈ s, HSMul.hSMul (α := ℝ) (Pi.single (M := fun (a : ℂ) ↦ ℝ) z 1 x : ℝ) (z - x)) 0
have pos₁ : ∀ b ∈ s, b ≠ z → HSMul.hSMul (α := ℝ) (Pi.single (M := fun (a : ℂ) ↦ ℝ) z 1 b : ℝ) (z - b) = 0
Tactic: rw [Finset.sum_eq_single z]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : eval z P = 0
pos : HSMul.hSMul (α := ℝ) (Pi.single (M := fun (a : ℂ) ↦ ℝ) z 1 z : ℝ) (z - z) = 0
pos₁ : ∀ b ∈ s, b ≠ z → HSMul.hSMul (α := ℝ) (Pi.single (M := fun (a : ℂ) ↦ ℝ) z 1 b : ℝ) (z - b) = 0
⊢ Eq (α := ℂ) (∑ x ∈ s, HSMul.hSMul (α := ℝ) (Pi.single (M := fun (a : ℂ) ↦ ℝ) z 1 x : ℝ) (z - x)) 0
have pos₂ : z ∉ s → HSMul.hSMul (α := ℝ) (Pi.single (M := fun (a : ℂ) ↦ ℝ) z 1 z : ℝ) (z - z) = 0
Tactic: rw [Finset.sum_eq_single z]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : ¬eval z P = 0
⊢ Eq (α := ℂ) (∑ x ∈ s, weight x • (z - x) : ℂ)
    ((starRingEnd ℂ : (a : ℂ) → ℂ) (∑ x ∈ s, rootMultiplicity x P • (1 / (z - x))) : ℂ)
have simp : Eq (α := ℂ) (∑ x ∈ s, ((↑(rootMultiplicity x P) : ℝ) / HPow.hPow (β := ℕ) ‖z - x‖ 2) • (z - x))
  (∑ x ∈ s, (starRingEnd ℂ : (a : ℂ) → ℂ) (rootMultiplicity x P • (1 / (z - x))))
Tactic: simp only [map_sum, weight, derivRootWeight, if_neg hzP]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : ¬eval z P = 0
⊢ Eq (α := ℂ) (∑ x ∈ s, ((↑(rootMultiplicity x P) : ℝ) / HPow.hPow (β := ℕ) ‖z - x‖ 2) • (z - x))
    (∑ x ∈ s, (starRingEnd ℂ : (a : ℂ) → ℂ) (rootMultiplicity x P • (1 / (z - x))))
have refine : ∀ (x : ℂ) (hx : x ∈ s),
  ((↑(rootMultiplicity x P) : ℝ) / HPow.hPow (β := ℕ) ‖z - x‖ 2) • (z - x) =
    (starRingEnd ℂ : (a : ℂ) → ℂ) (rootMultiplicity x P • (1 / (z - x)))
Tactic: refine Finset.sum_congr rfl fun x hx ↦ ?_
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : ¬eval z P = 0
x : ℂ
hx : x ∈ s
⊢ z - x ≠ 0
have rw : z ≠ x
Tactic: rw [sub_ne_zero]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (hypothesis)
Goal:
P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : ¬eval z P = 0
x : ℂ
hx : x ∈ s
⊢ ((↑(rootMultiplicity x P) : ℝ) / HPow.hPow (β := ℕ) ‖z - x‖ 2) • (z - x) =
    (starRingEnd ℂ : (a : ℂ) → ℂ) (rootMultiplicity x P • (1 / (z - x)))
have this : z - x ≠ 0
Tactic: have : z - x ≠ 0 := by
        rw [sub_ne_zero]
        rintro rfl
        simp_all [s]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : ¬eval z P = 0
⊢ Eq (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) (Multiset.sum (Multiset.map (fun (x : ℂ) ↦ 1 / (z - x)) P.roots)) : ℂ)
    (0 : ℂ)
have rw : Eq (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) (eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) / eval z P) : ℂ) (0 : ℂ)
Tactic: rw [← (IsAlgClosed.splits _).eval_derivative_div_eval_of_ne_zero hzP]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : ¬eval z P = 0
⊢ Eq (α := ℂ) (∑ x ∈ s, weight x • (z - x)) 0
have x : Eq (α := ℂ) (∑ x ∈ s, weight x • (z - x) : ℂ)
  ((starRingEnd ℂ : (a : ℂ) → ℂ) (∑ x ∈ s, rootMultiplicity x P • (1 / (z - x))) : ℂ)
Tactic: calc
    ∑ x ∈ s, weight x • (z - x) = conj (∑ x ∈ s, P.rootMultiplicity x • (1 / (z - x))) := by
      simp only [map_sum, weight, derivRootWeight, if_neg hzP]
      refine Finset.sum_congr rfl fun x hx ↦ ?_
      have : z - x ≠ 0 := by
        rw [sub_ne_zero]
        rintro rfl
        simp_all [s]
      simp [← Complex.conj_mul', field]
    _ = conj (P.roots.map fun x ↦ 1 / (z - x)).sum := by
      simp only [Finset.sum_multiset_map_count, P.count_roots, s]
    _ = 0 := by
      rw [← (IsAlgClosed.splits _).eval_derivative_div_eval_of_ne_zero hzP]
      simp [hz]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : ¬eval z P = 0
x :
  Eq (α := ℂ) (∑ x ∈ s, weight x • (z - x) : ℂ)
    ((starRingEnd ℂ : (a : ℂ) → ℂ) (∑ x ∈ s, rootMultiplicity x P • (1 / (z - x))) : ℂ)
⊢ Eq (α := ℂ) (∑ x ∈ s, weight x • (z - x)) 0
have x₁ : Eq (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) (∑ x ∈ s, rootMultiplicity x P • (1 / (z - x))) : ℂ)
  ((starRingEnd ℂ : (a : ℂ) → ℂ) (Multiset.sum (Multiset.map (fun (x : ℂ) ↦ 1 / (z - x)) P.roots)) : ℂ)
Tactic: calc
    ∑ x ∈ s, weight x • (z - x) = conj (∑ x ∈ s, P.rootMultiplicity x • (1 / (z - x))) := by
      simp only [map_sum, weight, derivRootWeight, if_neg hzP]
      refine Finset.sum_congr rfl fun x hx ↦ ?_
      have : z - x ≠ 0 := by
        rw [sub_ne_zero]
        rintro rfl
        simp_all [s]
      simp [← Complex.conj_mul', field]
    _ = conj (P.roots.map fun x ↦ 1 / (z - x)).sum := by
      simp only [Finset.sum_multiset_map_count, P.count_roots, s]
    _ = 0 := by
      rw [← (IsAlgClosed.splits _).eval_derivative_div_eval_of_ne_zero hzP]
      simp [hz]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
z : ℂ
hP : 0 < P.degree
hz : eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
weight : ℂ → ℝ := P.derivRootWeight z
s : Finset ℂ := P.roots.toFinset
hzP : ¬eval z P = 0
x :
  Eq (α := ℂ) (∑ x ∈ s, weight x • (z - x) : ℂ)
    ((starRingEnd ℂ : (a : ℂ) → ℂ) (∑ x ∈ s, rootMultiplicity x P • (1 / (z - x))) : ℂ)
x₁ :
  Eq (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) (∑ x ∈ s, rootMultiplicity x P • (1 / (z - x))) : ℂ)
    ((starRingEnd ℂ : (a : ℂ) → ℂ) (Multiset.sum (Multiset.map (fun (x : ℂ) ↦ 1 / (z - x)) P.roots)) : ℂ)
⊢ Eq (α := ℂ) (∑ x ∈ s, weight x • (z - x)) 0
have x₂ : Eq (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) (Multiset.sum (Multiset.map (fun (x : ℂ) ↦ 1 / (z - x)) P.roots)) : ℂ)
  (0 : ℂ)
Tactic: calc
    ∑ x ∈ s, weight x • (z - x) = conj (∑ x ∈ s, P.rootMultiplicity x • (1 / (z - x))) := by
      simp only [map_sum, weight, derivRootWeight, if_neg hzP]
      refine Finset.sum_congr rfl fun x hx ↦ ?_
      have : z - x ≠ 0 := by
        rw [sub_ne_zero]
        rintro rfl
        simp_all [s]
      simp [← Complex.conj_mul', field]
    _ = conj (P.roots.map fun x ↦ 1 / (z - x)).sum := by
      simp only [Finset.sum_multiset_map_count, P.count_roots, s]
    _ = 0 := by
      rw [← (IsAlgClosed.splits _).eval_derivative_div_eval_of_ne_zero hzP]
      simp [hz]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (hypothesis)
Goal:
P : ℂ[X]
h₀ : 0 < P.degree
z : ℂ
hz : z ∈ rootSet (T := ℂ) ((derivative : (a : ℂ[X]) → ℂ[X]) P) ℂ
⊢ Membership.mem (γ := Set ℂ) ((convexHull ℝ : (a : Set ℂ) → Set ℂ) (P.rootSet ℂ) : Set ℂ) z
have hz₁ : Ne (α := ℂ[X]) ((derivative : (a : ℂ[X]) → ℂ[X]) P : ℂ[X]) (0 : ℂ[X]) ∧ eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
Tactic: rw [mem_rootSet, coe_aeval_eq_eval] at hz
Cleared hypotheses: hz
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
h₀ : 0 < P.degree
z : ℂ
hz :
  Ne (α := ℂ[X]) ((derivative : (a : ℂ[X]) → ℂ[X]) P : ℂ[X]) (0 : ℂ[X]) ∧
    eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
⊢ Membership.mem (γ := Set ℂ) ((convexHull ℝ : (a : Set ℂ) → Set ℂ) (P.rootSet ℂ) : Set ℂ) z
have rw : Membership.mem (γ := Set ℂ) ((convexHull ℝ : (a : Set ℂ) → Set ℂ) (P.rootSet ℂ) : Set ℂ)
  (P.roots.toFinset.centerMass (P.derivRootWeight z) id)
Tactic: rw [eq_centerMass_of_eval_derivative_eq_zero h₀ hz.2]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
h₀ : 0 < P.degree
z : ℂ
hz :
  Ne (α := ℂ[X]) ((derivative : (a : ℂ[X]) → ℂ[X]) P : ℂ[X]) (0 : ℂ[X]) ∧
    eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
⊢ Membership.mem (γ := Set ℂ) ((convexHull ℝ : (a : Set ℂ) → Set ℂ) (P.rootSet ℂ) : Set ℂ)
    (P.roots.toFinset.centerMass (P.derivRootWeight z) id)
have hw₀ : ∀ i ∈ P.roots.toFinset, 0 ≤ P.derivRootWeight z i
Tactic: apply Finset.centerMass_mem_convexHull
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
h₀ : 0 < P.degree
z : ℂ
hz :
  Ne (α := ℂ[X]) ((derivative : (a : ℂ[X]) → ℂ[X]) P : ℂ[X]) (0 : ℂ[X]) ∧
    eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
hw₀ : ∀ i ∈ P.roots.toFinset, 0 ≤ P.derivRootWeight z i
⊢ Membership.mem (γ := Set ℂ) ((convexHull ℝ : (a : Set ℂ) → Set ℂ) (P.rootSet ℂ) : Set ℂ)
    (P.roots.toFinset.centerMass (P.derivRootWeight z) id)
have hws : LT.lt (α := ℝ) 0 (∑ i ∈ P.roots.toFinset, P.derivRootWeight z i)
Tactic: apply Finset.centerMass_mem_convexHull
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.GaussLucas (goal)
Goal:
P : ℂ[X]
h₀ : 0 < P.degree
z : ℂ
hz :
  Ne (α := ℂ[X]) ((derivative : (a : ℂ[X]) → ℂ[X]) P : ℂ[X]) (0 : ℂ[X]) ∧
    eval z ((derivative : (a : ℂ[X]) → ℂ[X]) P) = 0
hw₀ : ∀ i ∈ P.roots.toFinset, 0 ≤ P.derivRootWeight z i
hws : LT.lt (α := ℝ) 0 (∑ i ∈ P.roots.toFinset, P.derivRootWeight z i)
⊢ Membership.mem (γ := Set ℂ) ((convexHull ℝ : (a : Set ℂ) → Set ℂ) (P.rootSet ℂ) : Set ℂ)
    (P.roots.toFinset.centerMass (P.derivRootWeight z) id)
have hz₁ : ∀ i ∈ P.roots.toFinset, id i ∈ P.rootSet ℂ
Tactic: apply Finset.centerMass_mem_convexHull
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (goal)
Goal:
p : ℤ[X]
hp : p.IsUnitTrinomial
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p.mirror : ℂ) (0 : ℂ))
⊢ Irreducible p
have refine : ∀ (q : ℤ[X]) (hq : q ∣ p) (hq' : q ∣ p.mirror), IsUnit q
Tactic: refine hp.irreducible_of_coprime fun q hq hq' => ?_
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (goal)
Goal:
p : ℤ[X]
hp : p.IsUnitTrinomial
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p.mirror : ℂ) (0 : ℂ))
q : ℤ[X]
hq : q ∣ p
hq' : q ∣ p.mirror
this : ¬0 < q.natDegree
⊢ IsUnit q
have rcases : ∀ (p : ℤ[X]) (hp : (q * p).IsUnitTrinomial)
  (h :
    ∀ (z : ℂ),
      ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p) : ℂ) (0 : ℂ) ∧
          Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p).mirror : ℂ) (0 : ℂ)))
  (hq' : q ∣ (q * p).mirror), IsUnit q
Tactic: rcases hq with ⟨p, rfl⟩
Cleared hypotheses: p, hp, h, hq, hq'
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (hypothesis)
Goal:
q : ℤ[X]
this : ¬0 < q.natDegree
p : ℤ[X]
hp : (q * p).IsUnitTrinomial
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p) : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p).mirror : ℂ) (0 : ℂ))
hq' : q ∣ (q * p).mirror
⊢ IsUnit q
have hp₁ : IsUnit (q * p).leadingCoeff
Tactic: replace hp := hp.leadingCoeff_isUnit
Cleared hypotheses: hp
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (hypothesis)
Goal:
q : ℤ[X]
this : ¬0 < q.natDegree
p : ℤ[X]
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p) : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p).mirror : ℂ) (0 : ℂ))
hq' : q ∣ (q * p).mirror
hp : IsUnit (q * p).leadingCoeff
⊢ IsUnit q
have hp₁ : IsUnit (q.leadingCoeff * p.leadingCoeff)
Tactic: rw [leadingCoeff_mul] at hp
Cleared hypotheses: hp
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (hypothesis)
Goal:
q : ℤ[X]
this : ¬0 < q.natDegree
p : ℤ[X]
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p) : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p).mirror : ℂ) (0 : ℂ))
hq' : q ∣ (q * p).mirror
hp : IsUnit (q.leadingCoeff * p.leadingCoeff)
⊢ IsUnit q
have hp₁ : IsUnit q.leadingCoeff
Tactic: replace hp := isUnit_of_mul_isUnit_left hp
Cleared hypotheses: hp
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (goal)
Goal:
q : ℤ[X]
this : ¬0 < q.natDegree
p : ℤ[X]
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p) : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p).mirror : ℂ) (0 : ℂ))
hq' : q ∣ (q * p).mirror
hp : IsUnit q.leadingCoeff
⊢ IsUnit q
have rw : ∀ (this : q.natDegree = 0) (p : ℤ[X])
  (h :
    ∀ (z : ℂ),
      ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p) : ℂ) (0 : ℂ) ∧
          Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p).mirror : ℂ) (0 : ℂ)))
  (hq' : q ∣ (q * p).mirror) (hp : IsUnit q.leadingCoeff), IsUnit q
Tactic: rw [not_lt, Nat.le_zero] at this
Cleared hypotheses: this, p, h, hq', hp
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (goal)
Goal:
q : ℤ[X]
this : q.natDegree = 0
p : ℤ[X]
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p) : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p).mirror : ℂ) (0 : ℂ))
hq' : q ∣ (q * p).mirror
hp : IsUnit q.leadingCoeff
⊢ IsUnit q
have rwa : IsUnit (M := ℤ[X]) ((C : (a : ℤ) → ℤ[X]) (q.coeff 0) : ℤ[X])
Tactic: rwa [eq_C_of_natDegree_eq_zero this, isUnit_C, ← this]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (goal)
Goal:
q : ℤ[X]
this : q.natDegree = 0
p : ℤ[X]
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p) : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p).mirror : ℂ) (0 : ℂ))
hq' : q ∣ (q * p).mirror
hp : IsUnit q.leadingCoeff
rwa : IsUnit (M := ℤ[X]) ((C : (a : ℤ) → ℤ[X]) (q.coeff 0) : ℤ[X])
⊢ IsUnit q
have rwa₁ : IsUnit (q.coeff 0)
Tactic: rwa [eq_C_of_natDegree_eq_zero this, isUnit_C, ← this]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (goal)
Goal:
q : ℤ[X]
this : q.natDegree = 0
p : ℤ[X]
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p) : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * p).mirror : ℂ) (0 : ℂ))
hq' : q ∣ (q * p).mirror
hp : IsUnit q.leadingCoeff
rwa : IsUnit (M := ℤ[X]) ((C : (a : ℤ) → ℤ[X]) (q.coeff 0) : ℤ[X])
rwa₁ : IsUnit (q.coeff 0)
⊢ IsUnit q
have rwa₂ : IsUnit (q.coeff q.natDegree)
Tactic: rwa [eq_C_of_natDegree_eq_zero this, isUnit_C, ← this]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (goal)
Goal:
p : ℤ[X]
hp : p.IsUnitTrinomial
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p.mirror : ℂ) (0 : ℂ))
q : ℤ[X]
hq : q ∣ p
hq' : q ∣ p.mirror
⊢ IsUnit q
have suffices : ¬0 < q.natDegree
Tactic: suffices ¬0 < q.natDegree by
    rcases hq with ⟨p, rfl⟩
    replace hp := hp.leadingCoeff_isUnit
    rw [leadingCoeff_mul] at hp
    replace hp := isUnit_of_mul_isUnit_left hp
    rw [not_lt, Nat.le_zero] at this
    rwa [eq_C_of_natDegree_eq_zero this, isUnit_C, ← this]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (hypothesis)
Goal:
p : ℤ[X]
hp : p.IsUnitTrinomial
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p.mirror : ℂ) (0 : ℂ))
q : ℤ[X]
hq : q ∣ p
hq' : q ∣ p.mirror
hq'' : 0 < q.natDegree
⊢ False
have hq''₁ : 0 < q.degree
Tactic: rw [natDegree_pos_iff_degree_pos] at hq''
Cleared hypotheses: hq''
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (hypothesis)
Goal:
p : ℤ[X]
hp : p.IsUnitTrinomial
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p.mirror : ℂ) (0 : ℂ))
q : ℤ[X]
hq : q ∣ p
hq' : q ∣ p.mirror
hq'' : 0 < q.degree
⊢ False
have hq''₁ : 0 < (map (algebraMap ℤ ℂ) q).degree
Tactic: rw [← degree_map_eq_of_injective (algebraMap ℤ ℂ).injective_int] at hq''
Cleared hypotheses: hq''
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (goal)
Goal:
p : ℤ[X]
hp : p.IsUnitTrinomial
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p.mirror : ℂ) (0 : ℂ))
q : ℤ[X]
hq : q ∣ p
hq' : q ∣ p.mirror
hq'' : 0 < (map (algebraMap ℤ ℂ) q).degree
⊢ False
have obtain : ∀ (z : ℂ) (hz : (map (algebraMap ℤ ℂ) q).IsRoot z), False
Tactic: obtain ⟨z, hz⟩ := Complex.exists_root hq''
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (hypothesis)
Goal:
p : ℤ[X]
hp : p.IsUnitTrinomial
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p.mirror : ℂ) (0 : ℂ))
q : ℤ[X]
hq : q ∣ p
hq' : q ∣ p.mirror
hq'' : 0 < (map (algebraMap ℤ ℂ) q).degree
z : ℂ
hz : (map (algebraMap ℤ ℂ) q).IsRoot z
⊢ False
have hz₁ : Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) q : ℂ) (0 : ℂ)
Tactic: rw [IsRoot, eval_map_algebraMap] at hz
Cleared hypotheses: hz
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (goal)
Goal:
p : ℤ[X]
hp : p.IsUnitTrinomial
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p.mirror : ℂ) (0 : ℂ))
q : ℤ[X]
hq : q ∣ p
hq' : q ∣ p.mirror
hq'' : 0 < (map (algebraMap ℤ ℂ) q).degree
z : ℂ
hz : Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) q : ℂ) (0 : ℂ)
⊢ False
have refine_1 : Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p : ℂ) (0 : ℂ)
Tactic: refine h z ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (goal)
Goal:
p : ℤ[X]
hp : p.IsUnitTrinomial
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p.mirror : ℂ) (0 : ℂ))
q : ℤ[X]
hq : q ∣ p
hq' : q ∣ p.mirror
hq'' : 0 < (map (algebraMap ℤ ℂ) q).degree
z : ℂ
hz : Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) q : ℂ) (0 : ℂ)
refine_1 : Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p : ℂ) (0 : ℂ)
⊢ False
have refine_2 : Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p.mirror : ℂ) (0 : ℂ)
Tactic: refine h z ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (goal)
Goal:
p : ℤ[X]
hp : p.IsUnitTrinomial
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p.mirror : ℂ) (0 : ℂ))
q : ℤ[X]
hq : q ∣ p
hq' : q ∣ p.mirror
hq'' : 0 < (map (algebraMap ℤ ℂ) q).degree
z : ℂ
hz : Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) q : ℂ) (0 : ℂ)
⊢ Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p : ℂ) (0 : ℂ)
have refine_1 : ∀ (g' : ℤ[X]) (hg' : p = q * g'), Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p : ℂ) (0 : ℂ)
Tactic: obtain ⟨g', hg'⟩ := hq
Cleared hypotheses: hq
-- Lean 4 proof step from Mathlib.Analysis.Complex.Polynomial.UnitTrinomial (goal)
Goal:
p : ℤ[X]
hp : p.IsUnitTrinomial
h :
  ∀ (z : ℂ),
    ¬(Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p : ℂ) (0 : ℂ) ∧
        Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p.mirror : ℂ) (0 : ℂ))
q : ℤ[X]
hq' : q ∣ p.mirror
hq'' : 0 < (map (algebraMap ℤ ℂ) q).degree
z : ℂ
hz : Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) q : ℂ) (0 : ℂ)
g' : ℤ[X]
hg' : p = q * g'
⊢ Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) p : ℂ) (0 : ℂ)
have refine_1 : Eq (α := ℂ) ((aeval z : (a : ℤ[X]) → ℂ) (q * g') : ℂ) (0 : ℂ)
Tactic: rw [hg', aeval_mul, hz, zero_mul]