awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
a : R
inst : CommRing R
inst_1 : IsDomain R
inst_2 : DecidableEq R
p : R[X]
⊢ count a p.roots = rootMultiplicity a p
have pos : ∀ (hp : p = 0), count a p.roots = rootMultiplicity a p
Tactic: classical
  by_cases hp : p = 0
  · simp [hp]
  rw [roots_def, dif_neg hp]
  exact (Classical.choose_spec (exists_multiset_roots hp)).2 a
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
a : R
inst : CommRing R
inst_1 : IsDomain R
inst_2 : DecidableEq R
p : R[X]
pos : ∀ (hp : p = 0), count a p.roots = rootMultiplicity a p
⊢ count a p.roots = rootMultiplicity a p
have neg : ∀ (hp : ¬p = 0), count a p.roots = rootMultiplicity a p
Tactic: classical
  by_cases hp : p = 0
  · simp [hp]
  rw [roots_def, dif_neg hp]
  exact (Classical.choose_spec (exists_multiset_roots hp)).2 a
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
a : R
inst : CommRing R
inst_1 : IsDomain R
inst_2 : DecidableEq R
p : R[X]
pos : ∀ (hp : p = 0), count a p.roots = rootMultiplicity a p
neg neg₁ : ∀ (hp : ¬p = 0), count a p.roots = rootMultiplicity a p
⊢ count a p.roots = rootMultiplicity a p
have neg₂ : ∀ (hp : ¬p = 0),
  count a
      (Classical.choose (p := fun (x : Multiset R) ↦
        (↑x.card : WithBot ℕ) ≤ p.degree ∧ ∀ (a : R), count a x = rootMultiplicity a p) sorry) =
    rootMultiplicity a p
Tactic: classical
  by_cases hp : p = 0
  · simp [hp]
  rw [roots_def, dif_neg hp]
  exact (Classical.choose_spec (exists_multiset_roots hp)).2 a
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
a : R
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
⊢ a ∈ p.roots ↔ p ≠ 0 ∧ p.IsRoot a
have rw : 0 < count a p.roots ↔ p ≠ 0 ∧ p.IsRoot a
Tactic: rw [← count_pos, count_roots p, rootMultiplicity_pos']
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
a : R
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
rw : 0 < count a p.roots ↔ p ≠ 0 ∧ p.IsRoot a
⊢ a ∈ p.roots ↔ p ≠ 0 ∧ p.IsRoot a
have rw₁ : 0 < rootMultiplicity a p ↔ p ≠ 0 ∧ p.IsRoot a
Tactic: rw [← count_pos, count_roots p, rootMultiplicity_pos']
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
a : R
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
rw : 0 < count a p.roots ↔ p ≠ 0 ∧ p.IsRoot a
rw₁ : 0 < rootMultiplicity a p ↔ p ≠ 0 ∧ p.IsRoot a
⊢ a ∈ p.roots ↔ p ≠ 0 ∧ p.IsRoot a
have rw₂ : p ≠ 0 ∧ p.IsRoot a ↔ p ≠ 0 ∧ p.IsRoot a
Tactic: rw [← count_pos, count_roots p, rootMultiplicity_pos']
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
a : R
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
⊢ a ∈ p.roots ↔ p ≠ 0 ∧ p.IsRoot a
have rw : 0 < count a p.roots ↔ p ≠ 0 ∧ p.IsRoot a
Tactic: classical
  rw [← count_pos, count_roots p, rootMultiplicity_pos']
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
a : R
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
rw : 0 < count a p.roots ↔ p ≠ 0 ∧ p.IsRoot a
⊢ a ∈ p.roots ↔ p ≠ 0 ∧ p.IsRoot a
have rw₁ : 0 < rootMultiplicity a p ↔ p ≠ 0 ∧ p.IsRoot a
Tactic: classical
  rw [← count_pos, count_roots p, rootMultiplicity_pos']
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
a : R
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
rw : 0 < count a p.roots ↔ p ≠ 0 ∧ p.IsRoot a
rw₁ : 0 < rootMultiplicity a p ↔ p ≠ 0 ∧ p.IsRoot a
⊢ a ∈ p.roots ↔ p ≠ 0 ∧ p.IsRoot a
have rw₂ : p ≠ 0 ∧ p.IsRoot a ↔ p ≠ 0 ∧ p.IsRoot a
Tactic: classical
  rw [← count_pos, count_roots p, rootMultiplicity_pos']
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hp0 : p ≠ 0
⊢ p.roots = 0 ↔ p.IsRoot = ⊥
have refine : ∀ (h : p.roots = 0), p.IsRoot = ⊥
Tactic: refine ⟨fun h ↦ ?_, fun h ↦ eq_zero_of_forall_notMem fun x hx ↦ h ▸ mem_roots hp0 |>.mp hx⟩
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hp0 : p ≠ 0
h : p.roots = 0
⊢ p.IsRoot = ⊥
have h₁ : ∀ (a : R), p.IsRoot a ↔ ⊥ a
Tactic: ext a
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
⊢ p.roots = 0 ↔ p = 0 ∨ p.IsRoot = ⊥
have inl : Eq.{u + 1} (α := Multiset R) (roots 0) 0 ↔ Eq.{u + 1} (α := R[X]) 0 0 ∨ Eq.{u + 1} (α := (a : R) → Prop) (IsRoot 0) ⊥
Tactic: rcases eq_or_ne p 0 with rfl | hp0
Cleared hypotheses: p
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
inl :
  Eq.{u + 1} (α := Multiset R) (roots 0) 0 ↔ Eq.{u + 1} (α := R[X]) 0 0 ∨ Eq.{u + 1} (α := (a : R) → Prop) (IsRoot 0) ⊥
⊢ p.roots = 0 ↔ p = 0 ∨ p.IsRoot = ⊥
have inr : ∀ (hp0 : p ≠ 0), p.roots = 0 ↔ p = 0 ∨ p.IsRoot = ⊥
Tactic: rcases eq_or_ne p 0 with rfl | hp0
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
S : Type v
inst : CommRing R
inst_1 : IsDomain R
inst_2 : Semiring S
p : S[X]
f : S →+* R
hf : Function.Injective.{v + 1, u + 1} (α := S) (β := R) (⇑f : (a : S) → R)
x : R
hp : p ≠ 0
⊢ x ∈ (map f p).roots ↔ eval₂ f x p = 0
have rw : (map f p).IsRoot x ↔ eval₂ f x p = 0
Tactic: rw [mem_roots ((Polynomial.map_ne_zero_iff hf).mpr hp), IsRoot, eval_map]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
S : Type v
inst : CommRing R
inst_1 : IsDomain R
inst_2 : Semiring S
p : S[X]
f : S →+* R
hf : Function.Injective.{v + 1, u + 1} (α := S) (β := R) (⇑f : (a : S) → R)
x : R
hp : p ≠ 0
rw : (map f p).IsRoot x ↔ eval₂ f x p = 0
⊢ x ∈ (map f p).roots ↔ eval₂ f x p = 0
have rw₁ : eval x (map f p) = 0 ↔ eval₂ f x p = 0
Tactic: rw [mem_roots ((Polynomial.map_ne_zero_iff hf).mpr hp), IsRoot, eval_map]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
S : Type v
inst : CommRing R
inst_1 : IsDomain R
inst_2 : Semiring S
p : S[X]
f : S →+* R
hf : Function.Injective.{v + 1, u + 1} (α := S) (β := R) (⇑f : (a : S) → R)
x : R
hp : p ≠ 0
rw : (map f p).IsRoot x ↔ eval₂ f x p = 0
rw₁ : eval x (map f p) = 0 ↔ eval₂ f x p = 0
⊢ x ∈ (map f p).roots ↔ eval₂ f x p = 0
have rw₂ : eval₂ f x p = 0 ↔ eval₂ f x p = 0
Tactic: rw [mem_roots ((Polynomial.map_ne_zero_iff hf).mpr hp), IsRoot, eval_map]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
x : R
w : p ≠ 0
⊢ x ∈ p.roots ↔ Eq (α := R) ((aeval x : (a : R[X]) → R) p : R) (0 : R)
have rw : x ∈ p.roots ↔ eval₂ (algebraMap R R) x p = 0
Tactic: rw [aeval_def, ← mem_roots_map_of_injective (FaithfulSMul.algebraMap_injective _ _) w,
    Algebra.algebraMap_self, map_id]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
x : R
w : p ≠ 0
rw : x ∈ p.roots ↔ eval₂ (algebraMap R R) x p = 0
⊢ x ∈ p.roots ↔ Eq (α := R) ((aeval x : (a : R[X]) → R) p : R) (0 : R)
have rw₁ : x ∈ p.roots ↔ x ∈ roots (map (algebraMap R R) p)
Tactic: rw [aeval_def, ← mem_roots_map_of_injective (FaithfulSMul.algebraMap_injective _ _) w,
    Algebra.algebraMap_self, map_id]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
x : R
w : p ≠ 0
rw : x ∈ p.roots ↔ eval₂ (algebraMap R R) x p = 0
rw₁ : x ∈ p.roots ↔ x ∈ roots (map (algebraMap R R) p)
⊢ x ∈ p.roots ↔ Eq (α := R) ((aeval x : (a : R[X]) → R) p : R) (0 : R)
have rw₂ : x ∈ p.roots ↔ x ∈ (map (RingHom.id R) p).roots
Tactic: rw [aeval_def, ← mem_roots_map_of_injective (FaithfulSMul.algebraMap_injective _ _) w,
    Algebra.algebraMap_self, map_id]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
x : R
w : p ≠ 0
rw : x ∈ p.roots ↔ eval₂ (algebraMap R R) x p = 0
rw₁ : x ∈ p.roots ↔ x ∈ roots (map (algebraMap R R) p)
rw₂ : x ∈ p.roots ↔ x ∈ (map (RingHom.id R) p).roots
⊢ x ∈ p.roots ↔ Eq (α := R) ((aeval x : (a : R[X]) → R) p : R) (0 : R)
have rw₃ : x ∈ p.roots ↔ x ∈ p.roots
Tactic: rw [aeval_def, ← mem_roots_map_of_injective (FaithfulSMul.algebraMap_injective _ _) w,
    Algebra.algebraMap_self, map_id]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
h : Set.Infinite.{u} (α := R) {x : R | eval x p = eval x q}
⊢ p = q
have rw : p - q = 0
Tactic: rw [← sub_eq_zero]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
h : Set.Infinite.{u} (α := R) {x : R | eval x p = eval x q}
⊢ p - q = 0
have h₁ : Set.Infinite.{u} (α := R) {x : R | (p - q).IsRoot x}
Tactic: apply eq_zero_of_infinite_isRoot
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
hpq : p * q ≠ 0
r : R
⊢ count r (p * q).roots = count r (p.roots + q.roots)
have rw : count r (p * q).roots = count r p.roots + count r q.roots
Tactic: rw [count_add, count_roots, count_roots, count_roots, rootMultiplicity_mul hpq]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
hpq : p * q ≠ 0
r : R
rw : count r (p * q).roots = count r p.roots + count r q.roots
⊢ count r (p * q).roots = count r (p.roots + q.roots)
have rw₁ : rootMultiplicity r (p * q) = count r p.roots + count r q.roots
Tactic: rw [count_add, count_roots, count_roots, count_roots, rootMultiplicity_mul hpq]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
hpq : p * q ≠ 0
r : R
rw : count r (p * q).roots = count r p.roots + count r q.roots
rw₁ : rootMultiplicity r (p * q) = count r p.roots + count r q.roots
⊢ count r (p * q).roots = count r (p.roots + q.roots)
have rw₂ : rootMultiplicity r (p * q) = rootMultiplicity r p + count r q.roots
Tactic: rw [count_add, count_roots, count_roots, count_roots, rootMultiplicity_mul hpq]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
hpq : p * q ≠ 0
r : R
rw : count r (p * q).roots = count r p.roots + count r q.roots
rw₁ : rootMultiplicity r (p * q) = count r p.roots + count r q.roots
rw₂ : rootMultiplicity r (p * q) = rootMultiplicity r p + count r q.roots
⊢ count r (p * q).roots = count r (p.roots + q.roots)
have rw₃ : rootMultiplicity r (p * q) = rootMultiplicity r p + rootMultiplicity r q
Tactic: rw [count_add, count_roots, count_roots, count_roots, rootMultiplicity_mul hpq]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
hpq : p * q ≠ 0
r : R
rw : count r (p * q).roots = count r p.roots + count r q.roots
rw₁ : rootMultiplicity r (p * q) = count r p.roots + count r q.roots
rw₂ : rootMultiplicity r (p * q) = rootMultiplicity r p + count r q.roots
rw₃ : rootMultiplicity r (p * q) = rootMultiplicity r p + rootMultiplicity r q
⊢ count r (p * q).roots = count r (p.roots + q.roots)
have rw₄ : rootMultiplicity r p + rootMultiplicity r q = rootMultiplicity r p + rootMultiplicity r q
Tactic: rw [count_add, count_roots, count_roots, count_roots, rootMultiplicity_mul hpq]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
a x : R
⊢ x ∈ (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])).roots ↔ p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
have rw : HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X]) ≠ 0 ∧
    (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])).IsRoot x ↔
  p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
Tactic: rw [mem_roots', IsRoot.def, sub_ne_zero, eval_sub, sub_eq_zero, eval_C]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
a x : R
rw :
  HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X]) ≠ 0 ∧
      (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])).IsRoot x ↔
    p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
⊢ x ∈ (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])).roots ↔ p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
have rw₁ : HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X]) ≠ 0 ∧
    eval x (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])) = 0 ↔
  p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
Tactic: rw [mem_roots', IsRoot.def, sub_ne_zero, eval_sub, sub_eq_zero, eval_C]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
a x : R
rw :
  HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X]) ≠ 0 ∧
      (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])).IsRoot x ↔
    p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
rw₁ :
  HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X]) ≠ 0 ∧
      eval x (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])) = 0 ↔
    p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
⊢ x ∈ (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])).roots ↔ p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
have rw₂ : p ≠ (C : (a : R) → R[X]) a ∧ eval x (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])) = 0 ↔
  p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
Tactic: rw [mem_roots', IsRoot.def, sub_ne_zero, eval_sub, sub_eq_zero, eval_C]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
a x : R
rw :
  HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X]) ≠ 0 ∧
      (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])).IsRoot x ↔
    p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
rw₁ :
  HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X]) ≠ 0 ∧
      eval x (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])) = 0 ↔
    p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
rw₂ :
  p ≠ (C : (a : R) → R[X]) a ∧ eval x (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])) = 0 ↔
    p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
⊢ x ∈ (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])).roots ↔ p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
have rw₃ : p ≠ (C : (a : R) → R[X]) a ∧ eval x p - eval x ((C : (a : R) → R[X]) a) = 0 ↔ p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
Tactic: rw [mem_roots', IsRoot.def, sub_ne_zero, eval_sub, sub_eq_zero, eval_C]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
a x : R
rw :
  HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X]) ≠ 0 ∧
      (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])).IsRoot x ↔
    p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
rw₁ :
  HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X]) ≠ 0 ∧
      eval x (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])) = 0 ↔
    p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
rw₂ :
  p ≠ (C : (a : R) → R[X]) a ∧ eval x (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])) = 0 ↔
    p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
rw₃ :
  p ≠ (C : (a : R) → R[X]) a ∧ eval x p - eval x ((C : (a : R) → R[X]) a) = 0 ↔
    p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
⊢ x ∈ (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])).roots ↔ p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
have rw₄ : p ≠ (C : (a : R) → R[X]) a ∧ eval x p = eval x ((C : (a : R) → R[X]) a) ↔ p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
Tactic: rw [mem_roots', IsRoot.def, sub_ne_zero, eval_sub, sub_eq_zero, eval_C]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
a x : R
rw :
  HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X]) ≠ 0 ∧
      (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])).IsRoot x ↔
    p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
rw₁ :
  HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X]) ≠ 0 ∧
      eval x (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])) = 0 ↔
    p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
rw₂ :
  p ≠ (C : (a : R) → R[X]) a ∧ eval x (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])) = 0 ↔
    p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
rw₃ :
  p ≠ (C : (a : R) → R[X]) a ∧ eval x p - eval x ((C : (a : R) → R[X]) a) = 0 ↔
    p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
rw₄ :
  p ≠ (C : (a : R) → R[X]) a ∧ eval x p = eval x ((C : (a : R) → R[X]) a) ↔ p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
⊢ x ∈ (HSub.hSub (β := R[X]) p ((C : (a : R) → R[X]) a : R[X])).roots ↔ p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
have rw₅ : p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a ↔ p ≠ (C : (a : R) → R[X]) a ∧ eval x p = a
Tactic: rw [mem_roots', IsRoot.def, sub_ne_zero, eval_sub, sub_eq_zero, eval_C]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
r : R
⊢ (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) r : R[X])).roots = {r}
have a : ∀ (s : R), count s (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) r : R[X])).roots = count s {r}
Tactic: ext s
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
r s : R
⊢ count s (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) r : R[X])).roots = count s {r}
have a : rootMultiplicity s (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) r : R[X])) = count s {r}
Tactic: rw [count_roots, rootMultiplicity_X_sub_C, count_singleton]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
r s : R
a : rootMultiplicity s (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) r : R[X])) = count s {r}
⊢ count s (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) r : R[X])).roots = count s {r}
have a₁ : (if s = r then 1 else 0) = count s {r}
Tactic: rw [count_roots, rootMultiplicity_X_sub_C, count_singleton]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
r s : R
a : rootMultiplicity s (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) r : R[X])) = count s {r}
a₁ : (if s = r then 1 else 0) = count s {r}
⊢ count s (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) r : R[X])).roots = count s {r}
have a₂ : Eq (α := ℕ) (if s = r then 1 else 0) (if s = r then 1 else 0)
Tactic: rw [count_roots, rootMultiplicity_X_sub_C, count_singleton]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
r : R
⊢ (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) r : R[X])).roots = {r}
have a : ∀ (s : R), count s (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) r : R[X])).roots = count s {r}
Tactic: classical
  ext s
  rw [count_roots, rootMultiplicity_X_sub_C, count_singleton]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
⊢ Eq (α := Multiset R) (roots X) {0}
have rw : roots X = (HSub.hSub (α := R[X]) X ((C : (a : R) → R[X]) 0)).roots
Tactic: rw [← roots_X_sub_C, C_0, sub_zero]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
rw : roots X = (HSub.hSub (α := R[X]) X ((C : (a : R) → R[X]) 0)).roots
⊢ Eq (α := Multiset R) (roots X) {0}
have rw₁ : roots X = (HSub.hSub (α := R[X]) X 0).roots
Tactic: rw [← roots_X_sub_C, C_0, sub_zero]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
rw : roots X = (HSub.hSub (α := R[X]) X ((C : (a : R) → R[X]) 0)).roots
rw₁ : roots X = (HSub.hSub (α := R[X]) X 0).roots
⊢ Eq (α := Multiset R) (roots X) {0}
have rw₂ : Eq.{u + 1} (α := Multiset R) (roots X) (roots X)
Tactic: rw [← roots_X_sub_C, C_0, sub_zero]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
x : R
H : x = 0
⊢ Eq.{u + 1} (α := Multiset R) (roots ((C : (a : R) → R[X]) x)) 0
have rw : Eq.{u + 1} (α := Multiset R) (roots ((C : (a : R) → R[X]) 0)) 0
Tactic: rw [H, C_0, roots_zero]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
x : R
H : x = 0
rw : Eq.{u + 1} (α := Multiset R) (roots ((C : (a : R) → R[X]) 0)) 0
⊢ Eq.{u + 1} (α := Multiset R) (roots ((C : (a : R) → R[X]) x)) 0
have rw₁ : Eq.{u + 1} (α := Multiset R) (roots 0) 0
Tactic: rw [H, C_0, roots_zero]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
x : R
H : x = 0
rw : Eq.{u + 1} (α := Multiset R) (roots ((C : (a : R) → R[X]) 0)) 0
rw₁ : Eq.{u + 1} (α := Multiset R) (roots 0) 0
⊢ Eq.{u + 1} (α := Multiset R) (roots ((C : (a : R) → R[X]) x)) 0
have rw₂ : Eq.{u + 1} (α := Multiset R) 0 0
Tactic: rw [H, C_0, roots_zero]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
x : R
H : ¬x = 0
r : R
⊢ count r (roots ((C : (a : R) → R[X]) x)) = count r 0
have rw : rootMultiplicity r ((C : (a : R) → R[X]) x) = count r 0
Tactic: rw [count_roots, count_zero, rootMultiplicity_eq_zero (not_isRoot_C _ _ H)]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
x : R
H : ¬x = 0
r : R
rw : rootMultiplicity r ((C : (a : R) → R[X]) x) = count r 0
⊢ count r (roots ((C : (a : R) → R[X]) x)) = count r 0
have rw₁ : rootMultiplicity r ((C : (a : R) → R[X]) x) = 0
Tactic: rw [count_roots, count_zero, rootMultiplicity_eq_zero (not_isRoot_C _ _ H)]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
x : R
H : ¬x = 0
r : R
rw : rootMultiplicity r ((C : (a : R) → R[X]) x) = count r 0
rw₁ : rootMultiplicity r ((C : (a : R) → R[X]) x) = 0
⊢ count r (roots ((C : (a : R) → R[X]) x)) = count r 0
have rw₂ : Eq (α := ℕ) 0 0
Tactic: rw [count_roots, count_zero, rootMultiplicity_eq_zero (not_isRoot_C _ _ H)]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
a : R
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
ha : a ≠ 0
⊢ (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) a : R[X]) p).roots = p.roots
have pos : ∀ (hp : p = 0), (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) a : R[X]) p).roots = p.roots
Tactic: by_cases hp : p = 0
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
a : R
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
ha : a ≠ 0
pos : ∀ (hp : p = 0), (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) a : R[X]) p).roots = p.roots
⊢ (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) a : R[X]) p).roots = p.roots
have neg : ∀ (hp : ¬p = 0), (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) a : R[X]) p).roots = p.roots
Tactic: by_cases hp : p = 0
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
h : Associated p q
⊢ p.roots = q.roots
have obtain : ∀ (u : R[X]ˣ), p.roots = (p * (↑u : R[X])).roots
Tactic: obtain ⟨u, rfl⟩ := h
Cleared hypotheses: q, h
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
u : R[X]ˣ
⊢ p.roots = (p * (↑u : R[X])).roots
have rw : p.roots = (HMul.hMul (β := R[X]) p ((C : (a : R) → R[X]) ((↑u : R[X]).coeff 0) : R[X])).roots
Tactic: rw [eq_C_of_degree_eq_zero <| degree_coe_units u, mul_comm,
    roots_C_mul _ <| coeff_coe_units_zero_ne_zero u]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
u : R[X]ˣ
rw : p.roots = (HMul.hMul (β := R[X]) p ((C : (a : R) → R[X]) ((↑u : R[X]).coeff 0) : R[X])).roots
⊢ p.roots = (p * (↑u : R[X])).roots
have rw₁ : p.roots = (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑u : R[X]).coeff 0) : R[X]) p).roots
Tactic: rw [eq_C_of_degree_eq_zero <| degree_coe_units u, mul_comm,
    roots_C_mul _ <| coeff_coe_units_zero_ne_zero u]