awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
he :
  p =
    prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) * q
hq : q ≠ 0
right : p.roots.card + q.natDegree = p.natDegree
right₁ : q.roots = 0
⊢ p.roots.card + q.natDegree = p.natDegree ∧ q.roots = 0

Tactic:
constructor
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
he :
  p =
    prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) * q
hq : q ≠ 0
rw :
  (prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) *
      q).natDegree
| p.natDegree

Tactic:
rw [he]
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
he :
  p =
    prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) * q
hq : q ≠ 0
right :
  p.roots.card + q.natDegree =
    (prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) *
        q).natDegree
⊢ p.roots.card + q.natDegree = p.natDegree

Tactic:
conv_rhs => rw [he]
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
he :
  p =
    prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) * q
hq : q ≠ 0
right :
  p.roots.card + q.natDegree =
    natDegree (R := R)
        (prod
          (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots)) +
      q.natDegree
right₁ right₂ : p.roots.card + q.natDegree = p.roots.card + q.natDegree
⊢ p.roots.card + q.natDegree =
    (prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) *
        q).natDegree

Tactic:
rw [(monic_multisetProd_X_sub_C p.roots).natDegree_mul' hq,
      natDegree_multiset_prod_X_sub_C_eq_card]

Premises used: Polynomial.monic_multisetProd_X_sub_C, Polynomial.roots, Polynomial.natDegree_multiset_prod_X_sub_C_eq_card, Polynomial.Monic.natDegree_mul'
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
he :
  p =
    prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) * q
hq : q ≠ 0
⊢ (prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) *
        q).roots =
    p.roots

Tactic:
replace he := congr_arg roots he.symm
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
hq : q ≠ 0
he :
  (prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) *
        q).roots =
    p.roots
right : ∀ (he : p.roots + q.roots = p.roots), q.roots = 0
right₁ :
  prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) * q ≠
    0
⊢ q.roots = 0

Tactic:
rw [roots_mul, roots_multiset_prod_X_sub_C] at he

Premises used: Polynomial.roots_multiset_prod_X_sub_C, Polynomial.roots_mul
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
hq : q ≠ 0
he : p.roots + q.roots = p.roots
⊢ q.roots = 0

Tactic:
add_eq_left.1 he

Premises used: add_eq_left, Iff.mp
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
hq : q ≠ 0
he :
  (prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) *
        q).roots =
    p.roots
⊢ prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) * q ≠
    0

Tactic:
mul_ne_zero (monic_multisetProd_X_sub_C p.roots).ne_zero hq

Premises used: Polynomial.monic_multisetProd_X_sub_C, Polynomial.roots, mul_ne_zero, Polynomial.Monic.ne_zero
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
hq : q ≠ 0
he : p.roots + q.roots = p.roots
⊢ q.roots = 0

Tactic:
exacts [add_eq_left.1 he, mul_ne_zero (monic_multisetProd_X_sub_C p.roots).ne_zero hq]

Premises used: Polynomial.monic_multisetProd_X_sub_C, Polynomial.roots, add_eq_left, Iff.mp, mul_ne_zero, Polynomial.Monic.ne_zero
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hroots : p.roots.card = p.natDegree
⊢ HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
      (prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) :
        R[X]) =
    p

Tactic:
(eq_leadingCoeff_mul_of_monic_of_dvd_of_natDegree_le (monic_multisetProd_X_sub_C p.roots)
      p.prod_multiset_X_sub_C_dvd
      ((natDegree_multiset_prod_X_sub_C_eq_card _).trans hroots).ge).symm

Premises used: Eq.ge, Polynomial.monic_multisetProd_X_sub_C, Polynomial.roots, Polynomial.prod_multiset_X_sub_C_dvd, Polynomial.natDegree_multiset_prod_X_sub_C_eq_card, Eq.symm, Polynomial.eq_leadingCoeff_mul_of_monic_of_dvd_of_natDegree_le, Eq.trans
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hp : p.Monic
hroots : p.roots.card = p.natDegree
⊢ HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
      (prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) :
        R[X]) =
    p

Tactic:
C_leadingCoeff_mul_prod_multiset_X_sub_C hroots

Premises used: Polynomial.C_leadingCoeff_mul_prod_multiset_X_sub_C
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hp : p.Monic
hroots : p.roots.card = p.natDegree
h :
  prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) =
    HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
      (prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) :
        R[X])
⊢ prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) = p

Tactic:
convert! C_leadingCoeff_mul_prod_multiset_X_sub_C hroots

Premises used: Polynomial.C_leadingCoeff_mul_prod_multiset_X_sub_C
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hp : p.Monic
hroots : p.roots.card = p.natDegree
h :
  prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) =
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) 1)
      (prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots))
h₁ :
  prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) =
    HMul.hMul (α := R[X]) 1
      (prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots))
h₂ h₃ :
  Eq (α := R[X])
    (prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots))
    (prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots))
⊢ prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) =
    HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) p.leadingCoeff : R[X])
      (prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) p.roots) :
        R[X])

Tactic:
rw [hp.leadingCoeff, C_1, one_mul]

Premises used: Polynomial.C_1, Polynomial.Monic.leadingCoeff, one_mul
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
a p : R[X]
hp : p.Monic
hap : a ∣ p
⊢ a.leadingCoeff ∣ 1

Tactic:
simpa only [hp.leadingCoeff] using leadingCoeff_dvd_leadingCoeff hap

Premises used: Polynomial.leadingCoeff_dvd_leadingCoeff, Polynomial.Monic.leadingCoeff
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hirr : Irreducible p
inl : ∀ (hp : p.roots = 0), p.roots.card ≤ 1
inr : ∀ (x : R) (hx : x ∈ p.roots), p.roots.card ≤ 1
⊢ p.roots.card ≤ 1

Tactic:
obtain hp | ⟨x, hx⟩ := p.roots.empty_or_exists_mem

Premises used: Polynomial.roots, Multiset.empty_or_exists_mem
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hirr : Irreducible p
hp : p.roots = 0
⊢ p.roots.card ≤ 1

Tactic:
simp [hp]
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hirr : Irreducible p
x : R
hx : x ∈ p.roots
⊢ p.roots.card ≤ p.natDegree

Tactic:
p.card_roots'

Premises used: Polynomial.card_roots'
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hirr : Irreducible p
x : R
hx : x ∈ p.roots
h : 1 = p.natDegree
⊢ p.roots.card ≤ 1

Tactic:
convert! p.card_roots'

Premises used: Polynomial.card_roots'
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hirr : Irreducible p
x : R
hx : x ∈ p.roots
⊢ 1 = p.natDegree

Tactic:
(natDegree_eq_of_degree_eq_some <| degree_eq_one_of_irreducible_of_root hirr <|
    isRoot_of_mem_roots hx).symm

Premises used: Polynomial.natDegree_eq_of_degree_eq_some, Polynomial.isRoot_of_mem_roots, Polynomial.degree_eq_one_of_irreducible_of_root, Eq.symm
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hirr : Irreducible p
x : R
hx : x ∈ p.roots
⊢ 1 = p.natDegree

Tactic:
exact (natDegree_eq_of_degree_eq_some <| degree_eq_one_of_irreducible_of_root hirr <|
    isRoot_of_mem_roots hx).symm

Premises used: Polynomial.natDegree_eq_of_degree_eq_some, Polynomial.isRoot_of_mem_roots, Polynomial.degree_eq_one_of_irreducible_of_root, Eq.symm
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hirr : Irreducible p
hdeg : p.natDegree ≠ 1
by_contra : ∀ (hroots : ¬p.roots = 0), False
⊢ p.roots = 0

Tactic:
by_contra hroots
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hirr : Irreducible p
hdeg : p.natDegree ≠ 1
hroots : ¬p.roots = 0
have : ∀ (x : R) (hx : x ∈ p.roots), False
⊢ False

Tactic:
have ⟨x, hx⟩ := exists_mem_of_ne_zero hroots
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hirr : Irreducible p
hdeg : p.natDegree ≠ 1
hroots : ¬p.roots = 0
x : R
hx : x ∈ p.roots
⊢ False

Tactic:
hdeg <| natDegree_eq_of_degree_eq_some <|
    degree_eq_one_of_irreducible_of_root hirr (mem_roots'.mp hx).right

Premises used: Polynomial.natDegree_eq_of_degree_eq_some, And.right, Polynomial.mem_roots', Iff.mp, Polynomial.degree_eq_one_of_irreducible_of_root
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hirr : Irreducible p
hdeg : p.natDegree ≠ 1
hroots : ¬p.roots = 0
x : R
hx : x ∈ p.roots
⊢ False

Tactic:
exact hdeg <| natDegree_eq_of_degree_eq_some <|
    degree_eq_one_of_irreducible_of_root hirr (mem_roots'.mp hx).right

Premises used: Polynomial.natDegree_eq_of_degree_eq_some, And.right, Polynomial.mem_roots', Iff.mp, Polynomial.degree_eq_one_of_irreducible_of_root
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
p_monic : p.Monic
p_1 : p ≠ 1
simp :
  (∀ (q : R[X]), q.Monic → 0 < q.degree → q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → ¬q ∣ p) ↔
    ∀ (q : R[X]), q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → q ∣ p → IsUnit q
⊢ Irreducible p ↔ ∀ (q : R[X]), q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → q ∣ p → IsUnit q

Tactic:
simp only [p_monic.irreducible_iff_lt_natDegree_lt p_1, Finset.mem_Ioc, and_imp,
    natDegree_pos_iff_degree_pos, natDegree_le_iff_degree_le]

Premises used: and_imp, Polynomial.Monic.irreducible_iff_lt_natDegree_lt, Finset.mem_Ioc, Polynomial.natDegree_pos_iff_degree_pos, Polynomial.natDegree_le_iff_degree_le
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
p_monic : p.Monic
p_1 : p ≠ 1
mp :
  (∀ (q : R[X]), q.Monic → 0 < q.degree → q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → ¬q ∣ p) →
    ∀ (q : R[X]), q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → q ∣ p → IsUnit q
mpr :
  (∀ (q : R[X]), q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → q ∣ p → IsUnit q) →
    ∀ (q : R[X]), q.Monic → 0 < q.degree → q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → ¬q ∣ p
⊢ (∀ (q : R[X]), q.Monic → 0 < q.degree → q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → ¬q ∣ p) ↔
    ∀ (q : R[X]), q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → q ∣ p → IsUnit q

Tactic:
constructor
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
p_monic : p.Monic
p_1 : p ≠ 1
h : ∀ (q : R[X]), q.Monic → 0 < q.degree → q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → ¬q ∣ p
q : R[X]
deg_le : q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ)
dvd : q ∣ p
mp : ∀ (q_unit : ¬IsUnit q), False
⊢ IsUnit q

Tactic:
by_contra q_unit
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
p_monic : p.Monic
p_1 : p ≠ 1
h : ∀ (q : R[X]), q.Monic → 0 < q.degree → q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → ¬q ∣ p
q : R[X]
deg_le : q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ)
dvd : q ∣ p
q_unit : ¬IsUnit q
⊢ 0 < q.degree

Tactic:
have := degree_pos_of_not_isUnit_of_dvd_monic p_monic q_unit dvd
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
p_monic : p.Monic
p_1 : p ≠ 1
h : ∀ (q : R[X]), q.Monic → 0 < q.degree → q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → ¬q ∣ p
q : R[X]
deg_le : q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ)
dvd : q ∣ p
q_unit : ¬IsUnit q
this : 0 < q.degree
⊢ IsUnit q.leadingCoeff

Tactic:
have hu := p_monic.isUnit_leadingCoeff_of_dvd dvd
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
p_monic : p.Monic
p_1 : p ≠ 1
h : ∀ (q : R[X]), q.Monic → 0 < q.degree → q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → ¬q ∣ p
q : R[X]
deg_le : q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ)
dvd : q ∣ p
q_unit : ¬IsUnit q
this : 0 < q.degree
hu : IsUnit q.leadingCoeff
⊢ False

Tactic:
(h _ (monic_of_isUnit_leadingCoeff_inv_smul hu) ?_ ?_ (dvd_trans ?_ dvd)).elim

Premises used: Polynomial.monic_of_isUnit_leadingCoeff_inv_smul, False.elim, dvd_trans
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
p_monic : p.Monic
p_1 : p ≠ 1
h : ∀ (q : R[X]), q.Monic → 0 < q.degree → q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → ¬q ∣ p
q : R[X]
deg_le : q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ)
dvd : q ∣ p
q_unit : ¬IsUnit q
this : 0 < q.degree
hu : IsUnit q.leadingCoeff
mp : 0 < (hu.unit⁻¹ • q).degree
mp₁ : (hu.unit⁻¹ • q).degree ≤ (↑(p.natDegree / 2) : WithBot ℕ)
mp₂ : hu.unit⁻¹ • q ∣ q
⊢ False

Tactic:
refine (h _ (monic_of_isUnit_leadingCoeff_inv_smul hu) ?_ ?_ (dvd_trans ?_ dvd)).elim

Premises used: Polynomial.monic_of_isUnit_leadingCoeff_inv_smul, False.elim, dvd_trans
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
p_monic : p.Monic
p_1 : p ≠ 1
h : ∀ (q : R[X]), q.Monic → 0 < q.degree → q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → ¬q ∣ p
q : R[X]
deg_le : q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ)
dvd : q ∣ p
q_unit : ¬IsUnit q
this : 0 < q.degree
hu : IsUnit q.leadingCoeff
mp mp₁ mp₂ mp₃ : 0 < q.degree
⊢ 0 < (hu.unit⁻¹ • q).degree

Tactic:
rwa [degree_smul_of_smul_regular _ (isSMulRegular_of_group _)]

Premises used: Polynomial.degree_smul_of_smul_regular
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
p_monic : p.Monic
p_1 : p ≠ 1
h : ∀ (q : R[X]), q.Monic → 0 < q.degree → q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → ¬q ∣ p
q : R[X]
deg_le : q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ)
dvd : q ∣ p
q_unit : ¬IsUnit q
this : 0 < q.degree
hu : IsUnit q.leadingCoeff
mp mp₁ mp₂ mp₃ : q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ)
⊢ (hu.unit⁻¹ • q).degree ≤ (↑(p.natDegree / 2) : WithBot ℕ)

Tactic:
rwa [degree_smul_of_smul_regular _ (isSMulRegular_of_group _)]

Premises used: Polynomial.degree_smul_of_smul_regular
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
p_monic : p.Monic
p_1 : p ≠ 1
h : ∀ (q : R[X]), q.Monic → 0 < q.degree → q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → ¬q ∣ p
q : R[X]
deg_le : q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ)
dvd : q ∣ p
q_unit : ¬IsUnit q
this : 0 < q.degree
hu : IsUnit q.leadingCoeff
mp : (↑hu.unit⁻¹ : R) • q ∣ q
mp₁ : HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (↑hu.unit⁻¹ : R) : R[X]) q ∣ q
mp₂ mp₃ : q ∣ q
⊢ hu.unit⁻¹ • q ∣ q

Tactic:
rw [Units.smul_def, Polynomial.smul_eq_C_mul, (isUnit_C.mpr (Units.isUnit _)).mul_left_dvd]

Premises used: Polynomial.smul_eq_C_mul, IsUnit.mul_left_dvd, Units.smul_def
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
p_monic : p.Monic
p_1 : p ≠ 1
h : ∀ (q : R[X]), q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → q ∣ p → IsUnit q
q : R[X]
h_1 : q.Monic
deg_pos : 0 < q.degree
deg_le : q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ)
dvd : q ∣ p
⊢ False

Tactic:
deg_pos.ne' <| degree_eq_zero_of_isUnit (h q deg_le dvd)

Premises used: LT.lt.ne', Polynomial.degree_eq_zero_of_isUnit
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
p_monic : p.Monic
p_1 : p ≠ 1
h : ∀ (q : R[X]), q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ) → q ∣ p → IsUnit q
q : R[X]
h_1 : q.Monic
deg_pos : 0 < q.degree
deg_le : q.degree ≤ (↑(p.natDegree / 2) : WithBot ℕ)
dvd : q ∣ p
⊢ False

Tactic:
exact deg_pos.ne' <| degree_eq_zero_of_isUnit (h q deg_le dvd)

Premises used: LT.lt.ne', Polynomial.degree_eq_zero_of_isUnit
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
p : A[X]
f : A →+* B
hmap : map f p ≠ 0
a : A
rw :
  HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) ((f : (a : A) → B) a) : B[X]) ^ rootMultiplicity a p ∣
    map f p
⊢ rootMultiplicity a p ≤ rootMultiplicity ((f : (a : A) → B) a) (map f p)

Tactic:
rw [le_rootMultiplicity_iff hmap]

Premises used: Polynomial.le_rootMultiplicity_iff
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
p : A[X]
f : A →+* B
hmap : map f p ≠ 0
a : A
⊢ HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) ((f : (a : A) → B) a) : B[X]) ^ rootMultiplicity a p ∣
    map f p

Tactic:
_root_.trans ?_ (_root_.map_dvd (mapRingHom f) (pow_rootMultiplicity_dvd p a))

Premises used: Polynomial.pow_rootMultiplicity_dvd, trans, map_dvd, Polynomial.mapRingHom
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
p : A[X]
f : A →+* B
hmap : map f p ≠ 0
a : A
refine :
  HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) ((f : (a : A) → B) a) : B[X]) ^ rootMultiplicity a p ∣
    (mapRingHom f : (a : A[X]) → B[X])
      (HSub.hSub (α := A[X]) (β := A[X]) X ((C : (a : A) → A[X]) a : A[X]) ^ rootMultiplicity a p)
⊢ HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) ((f : (a : A) → B) a) : B[X]) ^ rootMultiplicity a p ∣
    map f p

Tactic:
refine _root_.trans ?_ (_root_.map_dvd (mapRingHom f) (pow_rootMultiplicity_dvd p a))

Premises used: Polynomial.pow_rootMultiplicity_dvd, trans, map_dvd, Polynomial.mapRingHom
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
p : A[X]
f : A →+* B
hmap : map f p ≠ 0
a : A
rw :
  HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) ((f : (a : A) → B) a) : B[X]) ^ rootMultiplicity a p ∣
    HPow.hPow (α := B[X])
      ((mapRingHom f : (a : A[X]) → B[X]) (HSub.hSub (α := A[X]) (β := A[X]) X ((C : (a : A) → A[X]) a : A[X])) : B[X])
      (rootMultiplicity a p)
rw₁ :
  HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) ((f : (a : A) → B) a) : B[X]) ^ rootMultiplicity a p ∣
    HSub.hSub (α := B[X]) (β := B[X]) ((mapRingHom f : (a : A[X]) → B[X]) X : B[X])
        ((mapRingHom f : (a : A[X]) → B[X]) ((C : (a : A) → A[X]) a) : B[X]) ^
      rootMultiplicity a p
rw₂ :
  HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) ((f : (a : A) → B) a) : B[X]) ^ rootMultiplicity a p ∣
    (map f X - map f ((C : (a : A) → A[X]) a)) ^ rootMultiplicity a p
rw₃ :
  HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) ((f : (a : A) → B) a) : B[X]) ^ rootMultiplicity a p ∣
    (X - map f ((C : (a : A) → A[X]) a)) ^ rootMultiplicity a p
rw₄ rw₅ :
  HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) ((f : (a : A) → B) a) : B[X]) ^ rootMultiplicity a p ∣
    HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) ((f : (a : A) → B) a) : B[X]) ^ rootMultiplicity a p
⊢ HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) ((f : (a : A) → B) a) : B[X]) ^ rootMultiplicity a p ∣
    (mapRingHom f : (a : A[X]) → B[X])
      (HSub.hSub (α := A[X]) (β := A[X]) X ((C : (a : A) → A[X]) a : A[X]) ^ rootMultiplicity a p)

Tactic:
rw [map_pow, map_sub, coe_mapRingHom, map_X, map_C]

Premises used: map_sub, Polynomial.map_X, map_pow, Polynomial.coe_mapRingHom, Polynomial.map_C
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
p : A[X]
f : A →+* B
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
a : A
pos : ∀ (hp0 : p = 0), rootMultiplicity a p = rootMultiplicity ((f : (a : A) → B) a) (map f p)
neg : ∀ (hp0 : ¬p = 0), rootMultiplicity a p = rootMultiplicity ((f : (a : A) → B) a) (map f p)
⊢ rootMultiplicity a p = rootMultiplicity ((f : (a : A) → B) a) (map f p)

Tactic:
by_cases hp0 : p = 0
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
p : A[X]
f : A →+* B
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
a : A
hp0 : p = 0
⊢ rootMultiplicity a p = rootMultiplicity ((f : (a : A) → B) a) (map f p)

Tactic:
simp only [hp0, rootMultiplicity_zero, Polynomial.map_zero]

Premises used: Polynomial.map_zero, Polynomial.rootMultiplicity_zero
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
p : A[X]
f : A →+* B
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
a : A
hp0 : ¬p = 0
neg : rootMultiplicity ((f : (a : A) → B) a) (map f p) ≤ rootMultiplicity a p
⊢ rootMultiplicity a p = rootMultiplicity ((f : (a : A) → B) a) (map f p)

Tactic:
apply le_antisymm (le_rootMultiplicity_map ((Polynomial.map_ne_zero_iff hf).mpr hp0) a)

Premises used: Iff.mpr, Polynomial.le_rootMultiplicity_map, le_antisymm, Polynomial.map_ne_zero_iff
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
p : A[X]
f : A →+* B
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
a : A
hp0 : ¬p = 0
neg :
  HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) ((f : (a : A) → B) a) : B[X]) ^
      rootMultiplicity ((f : (a : A) → B) a) (map f p) ∣
    map f p
⊢ rootMultiplicity ((f : (a : A) → B) a) (map f p) ≤ rootMultiplicity a p

Tactic:
rw [le_rootMultiplicity_iff hp0, ← map_dvd_map f hf ((monic_X_sub_C a).pow _),
    Polynomial.map_pow, Polynomial.map_sub, map_X, map_C]

Premises used: Polynomial.monic_X_sub_C, Polynomial.map_pow, Polynomial.map_X, Polynomial.le_rootMultiplicity_iff, Polynomial.Monic.pow, Polynomial.map_sub, Polynomial.map_dvd_map, Polynomial.map_C