awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
ι : Type u_2
inst_2 : Fintype ι
f : ι → R
hf : Function.Injective f
heval : ∀ (i : ι), eval (f i) p = 0
hcard : p.natDegree < Fintype.card ι
by_contra by_contra₁ by_contra₂ : ∀ (hp : ¬p = 0), False
by_contra₃ : ∀ (hp : ¬p = 0), #p.roots.toFinset < #p.roots.toFinset
by_contra₄ : ∀ (hp : ¬p = 0), image f univ ≤ p.roots.toFinset
by_contra₅ : ∀ (hp : ¬p = 0) ⦃x : R⦄, x ∈ image f univ → x ∈ p.roots.toFinset
by_contra₆ : ∀ (hp : ¬p = 0) ⦃x : R⦄ (x_1 : ι) (h : f x_1 = x), eval x p = 0
by_contra₇ : ∀ (hp : ¬p = 0) (x : ι), eval (f x) p = 0
⊢ p = 0

Tactic:
classical
  by_contra hp
  refine lt_irrefl #p.roots.toFinset ?_
  calc
    #p.roots.toFinset ≤ Multiset.card p.roots := Multiset.toFinset_card_le _
    _ ≤ natDegree p := Polynomial.card_roots' p
    _ < Fintype.card ι := hcard
    _ = Fintype.card (Set.range f) := (Set.card_range_of_injective hf).symm
    _ = #(Finset.univ.image f) := by rw [← Set.toFinset_card, Set.toFinset_range]
    _ ≤ #p.roots.toFinset := Finset.card_mono ?_
  intro _
  simp only [Finset.mem_image, Finset.mem_univ, true_and, Multiset.mem_toFinset, mem_roots', ne_eq,
    IsRoot.def, forall_exists_index, hp, not_false_eq_true]
  rintro x rfl
  exact heval _

Premises used: Finset.mem_univ, Polynomial.roots, Finset.mem_image, Multiset.toFinset_card_le, forall_exists_index, Polynomial.IsRoot.def, Fintype.card, ne_eq, Polynomial.card_roots', Polynomial.mem_roots', Multiset.card, Polynomial.natDegree, not_false_eq_true, Multiset.mem_toFinset, rfl, true_and, Finset.card_mono
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
ι : Type u_2
inst_2 : Fintype ι
f : ι → R
hf : Function.Injective f
heval : ∀ (i : ι), eval (f i) p = eval (f i) q
hcard : max p.natDegree q.natDegree < Fintype.card ι
rw : p - q = 0
⊢ p = q

Tactic:
rw [← sub_eq_zero]

Premises used: sub_eq_zero
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
ι : Type u_2
inst_2 : Fintype ι
f : ι → R
hf : Function.Injective f
heval : ∀ (i : ι), eval (f i) p = eval (f i) q
hcard : max p.natDegree q.natDegree < Fintype.card ι
heval₁ : ∀ (i : ι), eval (f i) (p - q) = 0
hcard₁ : (p - q).natDegree < Fintype.card ι
⊢ p - q = 0

Tactic:
apply eq_zero_of_natDegree_lt_card_of_eval_eq_zero _ hf

Premises used: Polynomial.eq_zero_of_natDegree_lt_card_of_eval_eq_zero
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
ι : Type u_2
inst_2 : Fintype ι
f : ι → R
hf : Function.Injective f
heval : ∀ (i : ι), eval (f i) p = eval (f i) q
hcard : max p.natDegree q.natDegree < Fintype.card ι
⊢ ∀ (i : ι), eval (f i) (p - q) = 0

Tactic:
simpa [sub_eq_zero]

Premises used: sub_eq_zero
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
ι : Type u_2
inst_2 : Fintype ι
f : ι → R
hf : Function.Injective f
heval : ∀ (i : ι), eval (f i) p = eval (f i) q
hcard : max p.natDegree q.natDegree < Fintype.card ι
⊢ (p - q).natDegree < Fintype.card ι

Tactic:
grind [natDegree_sub_le]

Premises used: Polynomial.natDegree_sub_le
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
s : Finset R
heval : ∀ i ∈ s, eval i p = 0
hcard : p.natDegree < #s
⊢ p = 0

Tactic:
eq_zero_of_natDegree_lt_card_of_eval_eq_zero p Subtype.val_injective
    (fun i : s ↦ heval i i.prop) (hcard.trans_eq (Fintype.card_coe s).symm)

Premises used: Subtype.val_injective, LT.lt.trans_eq, Polynomial.eq_zero_of_natDegree_lt_card_of_eval_eq_zero, Subtype.prop, Eq.symm, Subtype.val, Fintype.card_coe
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
p q : R[X]
s : Finset R
heval : ∀ i ∈ s, eval i p = eval i q
hcard : max p.natDegree q.natDegree < #s
⊢ p = q

Tactic:
eq_of_natDegree_lt_card_of_eval_eq p q Subtype.val_injective
    (fun i : s ↦ heval i i.prop) (hcard.trans_eq (Fintype.card_coe s).symm)

Premises used: Subtype.val_injective, Polynomial.eq_of_natDegree_lt_card_of_eval_eq, LT.lt.trans_eq, Subtype.prop, Eq.symm, Subtype.val, Fintype.card_coe
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
f : R[X]
hf : ∀ (r : R), eval r f = 0
hfR : (↑f.natDegree : Cardinal.{u}) < #R
inl : ∀ (hR : Finite R), f = 0
inr : ∀ (hR : Infinite R), f = 0
⊢ f = 0

Tactic:
obtain hR | hR := finite_or_infinite R

Premises used: finite_or_infinite
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
f : R[X]
hf : ∀ (r : R), eval r f = 0
hfR : (↑f.natDegree : Cardinal.{u}) < #R
hR : Finite R
inl : ∀ (this : Fintype R), f = 0
⊢ f = 0

Tactic:
have := Fintype.ofFinite R
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
f : R[X]
hf : ∀ (r : R), eval r f = 0
hfR : (↑f.natDegree : Cardinal.{u}) < #R
hR : Finite R
this : Fintype R
inl : f.natDegree < Fintype.card R
⊢ f = 0

Tactic:
apply eq_zero_of_natDegree_lt_card_of_eval_eq_zero f Function.injective_id hf

Premises used: Function.injective_id, Polynomial.eq_zero_of_natDegree_lt_card_of_eval_eq_zero
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
f : R[X]
hf : ∀ (r : R), eval r f = 0
hfR : (↑f.natDegree : Cardinal.{u}) < #R
hR : Finite R
this : Fintype R
⊢ f.natDegree < Fintype.card R

Tactic:
simpa only [mk_fintype, Nat.cast_lt] using hfR

Premises used: Cardinal.mk_fintype, Nat.cast_lt
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
f : R[X]
hf : ∀ (r : R), eval r f = 0
hfR : (↑f.natDegree : Cardinal.{u}) < #R
hR : Infinite R
⊢ f = 0

Tactic:
zero_of_eval_zero _ hf

Premises used: Polynomial.zero_of_eval_zero
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
f : R[X]
hf : ∀ (r : R), eval r f = 0
hfR : (↑f.natDegree : Cardinal.{u}) < #R
hR : Infinite R
⊢ f = 0

Tactic:
exact zero_of_eval_zero _ hf

Premises used: Polynomial.zero_of_eval_zero
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
f : R[X]
hf : f ≠ 0
hfR : (↑f.natDegree : Cardinal.{u}) < #R
contrapose! : ∀ (hf : ∀ (r : R), eval r f = 0), f = 0
⊢ ∃ (r : R), eval r f ≠ 0

Tactic:
contrapose! hf
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
f : R[X]
hfR : (↑f.natDegree : Cardinal.{u}) < #R
hf : ∀ (r : R), eval r f = 0
⊢ f = 0

Tactic:
eq_zero_of_forall_eval_zero_of_natDegree_lt_card f hf hfR

Premises used: Polynomial.eq_zero_of_forall_eval_zero_of_natDegree_lt_card
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
f : R[X]
hfR : (↑f.natDegree : Cardinal.{u}) < #R
hf : ∀ (r : R), eval r f = 0
⊢ f = 0

Tactic:
exact eq_zero_of_forall_eval_zero_of_natDegree_lt_card f hf hfR

Premises used: Polynomial.eq_zero_of_forall_eval_zero_of_natDegree_lt_card
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
s : Multiset R
⊢ Monic (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])) s))

Tactic:
monic_multiset_prod_of_monic _ _ fun a _ => monic_X_sub_C a

Premises used: Polynomial.monic_X_sub_C, Polynomial.monic_multiset_prod_of_monic
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
α : Type u_1
b : α → R
s : Finset α
⊢ Monic.{u} (R := R) (∏ a ∈ s, HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) (b a) : R[X]))

Tactic:
monic_prod_of_monic _ _ fun a _ => monic_X_sub_C (b a)

Premises used: Polynomial.monic_X_sub_C, Polynomial.monic_prod_of_monic
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
α : Type u_1
b : α → R
⊢ Monic.{u} (R := R) (∏ᶠ (k : α), HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) (b k) : R[X]))

Tactic:
monic_finprod_of_monic _ _ fun a _ => monic_X_sub_C (b a)

Premises used: Polynomial.monic_X_sub_C, Polynomial.monic_finprod_of_monic
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
inst_2 : DecidableEq R
⊢ 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))
    (∏ a ∈ p.roots.toFinset, HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X]) ^ rootMultiplicity a p)

Tactic:
simp only [count_roots, Finset.prod_multiset_map_count]

Premises used: Finset.prod_multiset_map_count, Polynomial.count_roots
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
rw :
  Dvd.dvd (α := (FractionRing R)[X])
    (map (algebraMap R (FractionRing R) : R →+* FractionRing 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)))
    (map (algebraMap R (FractionRing R)) 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) ∣ p

Tactic:
rw [← map_dvd_map _ (IsFractionRing.injective R <| FractionRing R)
    (monic_multisetProd_X_sub_C p.roots)]

Premises used: Polynomial.monic_multisetProd_X_sub_C, Polynomial.roots, IsFractionRing.injective, FractionRing, Polynomial.map_dvd_map
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
rw :
  Dvd.dvd (α := (FractionRing R)[X])
    (∏ i ∈ p.roots.toFinset,
      map (algebraMap R (FractionRing R))
        (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) i : R[X]) ^ rootMultiplicity i p))
    (map (algebraMap R (FractionRing R)) p)
⊢ Dvd.dvd (α := (FractionRing R)[X])
    (map (algebraMap R (FractionRing R) : R →+* FractionRing 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)))
    (map (algebraMap R (FractionRing R)) p)

Tactic:
rw [prod_multiset_root_eq_finset_root, Polynomial.map_prod]

Premises used: Polynomial.prod_multiset_root_eq_finset_root, Polynomial.map_prod
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
⊢ Dvd.dvd (α := (FractionRing R)[X])
    (∏ i ∈ p.roots.toFinset,
      map (algebraMap R (FractionRing R))
        (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) i : R[X]) ^ rootMultiplicity i p))
    (map (algebraMap R (FractionRing R)) p)

Tactic:
Finset.prod_dvd_of_coprime (fun a _ b _ h => ?_) fun a _ => ?_

Premises used: Finset.prod_dvd_of_coprime
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
refine_1 :
  ∀ a ∈ (↑p.roots.toFinset : Set R),
    ∀ b ∈ (↑p.roots.toFinset : Set R),
      ∀ (h : a ≠ b),
        Function.onFun (β := (FractionRing R)[X]) IsCoprime
          (fun (i : R) ↦
            map (algebraMap R (FractionRing R))
              (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) i : R[X]) ^ rootMultiplicity i p))
          a b
refine_2 :
  ∀ a ∈ p.roots.toFinset,
    Dvd.dvd (α := (FractionRing R)[X])
      (map (algebraMap R (FractionRing R))
        (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X]) ^ rootMultiplicity a p))
      (map (algebraMap R (FractionRing R)) p)
⊢ Dvd.dvd (α := (FractionRing R)[X])
    (∏ i ∈ p.roots.toFinset,
      map (algebraMap R (FractionRing R))
        (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) i : R[X]) ^ rootMultiplicity i p))
    (map (algebraMap R (FractionRing R)) p)

Tactic:
refine Finset.prod_dvd_of_coprime (fun a _ b _ h => ?_) fun a _ => ?_

Premises used: Finset.prod_dvd_of_coprime
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
a : R
h_1 : a ∈ (↑p.roots.toFinset : Set R)
b : R
h_2 : b ∈ (↑p.roots.toFinset : Set R)
h : a ≠ b
refine_1 :
  Function.onFun (β := (FractionRing R)[X]) IsCoprime
    (fun (i : R) ↦
      HSub.hSub (α := (FractionRing R)[X]) (β := (FractionRing R)[X]) X
          ((C : (a : FractionRing R) → (FractionRing R)[X])
              (((algebraMap R (FractionRing R) : R →+* FractionRing R) : (a : R) → FractionRing R) i) :
            (FractionRing R)[X]) ^
        rootMultiplicity i p)
    a b
⊢ Function.onFun (β := (FractionRing R)[X]) IsCoprime
    (fun (i : R) ↦
      map (algebraMap R (FractionRing R))
        (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) i : R[X]) ^ rootMultiplicity i p))
    a b

Tactic:
simp_rw [Polynomial.map_pow, Polynomial.map_sub, map_C, map_X]

Premises used: Polynomial.map_pow, Polynomial.map_X, Polynomial.map_sub, Polynomial.map_C
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
a : R
h_1 : a ∈ (↑p.roots.toFinset : Set R)
b : R
h_2 : b ∈ (↑p.roots.toFinset : Set R)
h : a ≠ b
⊢ Function.onFun (β := (FractionRing R)[X]) IsCoprime
    (fun (i : R) ↦
      HSub.hSub (α := (FractionRing R)[X]) (β := (FractionRing R)[X]) X
          ((C : (a : FractionRing R) → (FractionRing R)[X])
              (((algebraMap R (FractionRing R) : R →+* FractionRing R) : (a : R) → FractionRing R) i) :
            (FractionRing R)[X]) ^
        rootMultiplicity i p)
    a b

Tactic:
(pairwise_coprime_X_sub_C (IsFractionRing.injective R <| FractionRing R) h).pow

Premises used: IsFractionRing.injective, FractionRing, IsCoprime.pow, Polynomial.pairwise_coprime_X_sub_C
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
a : R
h_1 : a ∈ (↑p.roots.toFinset : Set R)
b : R
h_2 : b ∈ (↑p.roots.toFinset : Set R)
h : a ≠ b
⊢ Function.onFun (β := (FractionRing R)[X]) IsCoprime
    (fun (i : R) ↦
      HSub.hSub (α := (FractionRing R)[X]) (β := (FractionRing R)[X]) X
          ((C : (a : FractionRing R) → (FractionRing R)[X])
              (((algebraMap R (FractionRing R) : R →+* FractionRing R) : (a : R) → FractionRing R) i) :
            (FractionRing R)[X]) ^
        rootMultiplicity i p)
    a b

Tactic:
exact (pairwise_coprime_X_sub_C (IsFractionRing.injective R <| FractionRing R) h).pow

Premises used: IsFractionRing.injective, FractionRing, IsCoprime.pow, Polynomial.pairwise_coprime_X_sub_C
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
a : R
h : a ∈ p.roots.toFinset
⊢ Dvd.dvd (α := (FractionRing R)[X])
    (map (algebraMap R (FractionRing R))
      (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X]) ^ rootMultiplicity a p))
    (map (algebraMap R (FractionRing R)) p)

Tactic:
Polynomial.map_dvd _ (pow_rootMultiplicity_dvd p a)

Premises used: Polynomial.pow_rootMultiplicity_dvd, Polynomial.map_dvd
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
a : R
h : a ∈ p.roots.toFinset
⊢ Dvd.dvd (α := (FractionRing R)[X])
    (map (algebraMap R (FractionRing R))
      (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X]) ^ rootMultiplicity a p))
    (map (algebraMap R (FractionRing R)) p)

Tactic:
exact Polynomial.map_dvd _ (pow_rootMultiplicity_dvd p a)

Premises used: Polynomial.pow_rootMultiplicity_dvd, Polynomial.map_dvd
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
rw rw₁ rw₂ :
  Dvd.dvd (α := (FractionRing R)[X])
    (map (algebraMap R (FractionRing R) : R →+* FractionRing 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)))
    (map (algebraMap R (FractionRing R)) p)
rw₃ :
  Dvd.dvd (α := (FractionRing R)[X])
    (∏ i ∈ p.roots.toFinset,
      map (algebraMap R (FractionRing R))
        (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) i : R[X]) ^ rootMultiplicity i p))
    (map (algebraMap R (FractionRing R)) p)
refine_1 :
  ∀ a ∈ (↑p.roots.toFinset : Set R),
    ∀ b ∈ (↑p.roots.toFinset : Set R),
      ∀ (h : a ≠ b),
        Function.onFun (β := (FractionRing R)[X]) IsCoprime
          (fun (i : R) ↦
            map (algebraMap R (FractionRing R))
              (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) i : R[X]) ^ rootMultiplicity i p))
          a b
refine_2 refine_2₁ :
  ∀ a ∈ p.roots.toFinset,
    Dvd.dvd (α := (FractionRing R)[X])
      (map (algebraMap R (FractionRing R))
        (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X]) ^ rootMultiplicity a p))
      (map (algebraMap R (FractionRing R)) 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) ∣ p

Tactic:
classical
  rw [← map_dvd_map _ (IsFractionRing.injective R <| FractionRing R)
    (monic_multisetProd_X_sub_C p.roots)]
  rw [prod_multiset_root_eq_finset_root, Polynomial.map_prod]
  refine Finset.prod_dvd_of_coprime (fun a _ b _ h => ?_) fun a _ => ?_
  · simp_rw [Polynomial.map_pow, Polynomial.map_sub, map_C, map_X]
    exact (pairwise_coprime_X_sub_C (IsFractionRing.injective R <| FractionRing R) h).pow
  · exact Polynomial.map_dvd _ (pow_rootMultiplicity_dvd p a)

Premises used: Polynomial.monic_multisetProd_X_sub_C, Polynomial.roots, IsFractionRing.injective, Polynomial.prod_multiset_root_eq_finset_root, FractionRing, Polynomial.map_pow, Polynomial.map_X, Polynomial.map_prod, Polynomial.map_sub, Polynomial.map_dvd_map, Polynomial.map_C, Finset.prod_dvd_of_coprime
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hp : p ≠ 0
s : Multiset R
h : prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) s) ∣ p
r : R
rw :
  prod
      (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X]))
        (Multiset.filter (Eq r) s)) ∣
    p
⊢ count r s ≤ count r p.roots

Tactic:
rw [count_roots, le_rootMultiplicity_iff hp, ← Multiset.prod_replicate, ←
        Multiset.map_replicate fun a => X - C a, ← Multiset.filter_eq]

Premises used: Multiset.prod_replicate, Polynomial.C, HSub.hSub, Polynomial.le_rootMultiplicity_iff, Polynomial.count_roots, Multiset.map_replicate, Multiset.filter_eq, Polynomial.X, DFunLike.coe
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hp : p ≠ 0
s : Multiset R
h : prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) s) ∣ p
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]))
        (Multiset.filter (Eq r) s)) ∣
    p

Tactic:
(Multiset.prod_dvd_prod_of_le <| Multiset.map_le_map <| s.filter_le _).trans h

Premises used: Multiset.prod_dvd_prod_of_le, Multiset.map_le_map, Multiset.filter_le, Dvd.dvd.trans
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hp : p ≠ 0
s : Multiset R
h : prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) s) ∣ p
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]))
        (Multiset.filter (Eq r) s)) ∣
    p

Tactic:
exact (Multiset.prod_dvd_prod_of_le <| Multiset.map_le_map <| s.filter_le _).trans h

Premises used: Multiset.prod_dvd_prod_of_le, Multiset.map_le_map, Multiset.filter_le, Dvd.dvd.trans
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hp : p ≠ 0
s : Multiset R
⊢ prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) s) ∣ p ↔
    s ≤ p.roots

Tactic:
exact
  ⟨fun h =>
    Multiset.le_iff_count.2 fun r => by
      rw [count_roots, le_rootMultiplicity_iff hp, ← Multiset.prod_replicate, ←
        Multiset.map_replicate fun a => X - C a, ← Multiset.filter_eq]
      exact (Multiset.prod_dvd_prod_of_le <| Multiset.map_le_map <| s.filter_le _).trans h,
    fun h =>
    (Multiset.prod_dvd_prod_of_le <| Multiset.map_le_map h).trans p.prod_multiset_X_sub_C_dvd⟩
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
hp : p ≠ 0
s : Multiset R
⊢ prod (Multiset.map (fun (a : R) ↦ HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) s) ∣ p ↔
    s ≤ p.roots

Tactic:
classical exact
  ⟨fun h =>
    Multiset.le_iff_count.2 fun r => by
      rw [count_roots, le_rootMultiplicity_iff hp, ← Multiset.prod_replicate, ←
        Multiset.map_replicate fun a => X - C a, ← Multiset.filter_eq]
      exact (Multiset.prod_dvd_prod_of_le <| Multiset.map_le_map <| s.filter_le _).trans h,
    fun h =>
    (Multiset.prod_dvd_prod_of_le <| Multiset.map_le_map h).trans p.prod_multiset_X_sub_C_dvd⟩
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
obtain :
  ∀ (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),
    ∃ (q : 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) *
            q =
          p ∧
        p.roots.card + q.natDegree = p.natDegree ∧ q.roots = 0
⊢ ∃ (q : 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) *
          q =
        p ∧
      p.roots.card + q.natDegree = p.natDegree ∧ q.roots = 0

Tactic:
obtain ⟨q, he⟩ := p.prod_multiset_X_sub_C_dvd

Premises used: Polynomial.prod_multiset_X_sub_C_dvd
-- 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
right : p.roots.card + q.natDegree = p.natDegree ∧ q.roots = 0
⊢ ∃ (q : 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) *
          q =
        p ∧
      p.roots.card + q.natDegree = p.natDegree ∧ q.roots = 0

Tactic:
use q, he.symm

Premises used: Eq.symm
-- 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
right :
  ∀
    (he :
      p =
        HMul.hMul (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))
          0),
    p.roots.card + natDegree.{u} (R := R) 0 = p.natDegree ∧ Eq.{u + 1} (α := Multiset R) (roots 0) 0
right₁ : ∀ (hq : q ≠ 0), p.roots.card + q.natDegree = p.natDegree ∧ q.roots = 0
⊢ p.roots.card + q.natDegree = p.natDegree ∧ q.roots = 0

Tactic:
obtain rfl | hq := eq_or_ne q 0

Premises used: eq_or_ne, OfNat.ofNat, rfl
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
he :
  p =
    HMul.hMul (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))
      0
⊢ p = 0

Tactic:
rw [mul_zero] at he

Premises used: MulZeroClass.mul_zero
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
p : R[X]
he : p = 0
right :
  Multiset.card.{u} (α := R) (roots 0) + natDegree.{u} (R := R) 0 = natDegree.{u} (R := R) 0 ∧
    Eq.{u + 1} (α := Multiset R) (roots 0) 0
⊢ p.roots.card + natDegree.{u} (R := R) 0 = p.natDegree ∧ Eq.{u + 1} (α := Multiset R) (roots 0) 0

Tactic:
subst he
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
R : Type u
inst : CommRing R
inst_1 : IsDomain R
⊢ Multiset.card.{u} (α := R) (roots 0) + natDegree.{u} (R := R) 0 = natDegree.{u} (R := R) 0 ∧
    Eq.{u + 1} (α := Multiset R) (roots 0) 0

Tactic:
simp