awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
⊢ 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

Tactic:
apply pow_rootMultiplicity_dvd

Premises used: Polynomial.pow_rootMultiplicity_dvd
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : DecidableEq B
p : A[X]
f : A →+* B
hmap : map f p ≠ 0
b : B
rw :
  prod
      (Multiset.map (fun (a : B) ↦ HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) a : B[X]))
        (replicate (count b (Multiset.map (⇑f : (a : A) → B) p.roots)) b)) ∣
    map f p
⊢ count b (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ rootMultiplicity b (map f p)

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

Premises used: Multiset.prod_replicate, Polynomial.C, HSub.hSub, Polynomial.le_rootMultiplicity_iff, Multiset.map_replicate, Polynomial.X, DFunLike.coe
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : DecidableEq B
p : A[X]
f : A →+* B
hmap : map f p ≠ 0
b : B
rw :
  prod
      (Multiset.map (fun (a : B) ↦ HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) a : B[X]))
        (Multiset.filter (Eq b) (Multiset.map (⇑f : (a : A) → B) p.roots))) ∣
    map f p
⊢ prod
      (Multiset.map (fun (a : B) ↦ HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) a : B[X]))
        (replicate (count b (Multiset.map (⇑f : (a : A) → B) p.roots)) b)) ∣
    map f p

Tactic:
rw [← Multiset.filter_eq]

Premises used: Multiset.filter_eq
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : DecidableEq B
p : A[X]
f : A →+* B
hmap : map f p ≠ 0
b : B
⊢ prod
      (Multiset.map (fun (a : B) ↦ HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) a : B[X]))
        (Multiset.filter (Eq b) (Multiset.map (⇑f : (a : A) → B) p.roots))) ∣
    map f p

Tactic:
(Multiset.prod_dvd_prod_of_le <| Multiset.map_le_map <| Multiset.filter_le (Eq b) _).trans ?_

Premises used: Multiset.prod_dvd_prod_of_le, Multiset.map_le_map, Multiset.filter_le, Eq, Dvd.dvd.trans
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : DecidableEq B
p : A[X]
f : A →+* B
hmap : map f p ≠ 0
b : B
this :
  prod
      (Multiset.map (fun (a : B) ↦ HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) a : B[X]))
        (Multiset.map (⇑f : (a : A) → B) p.roots)) ∣
    map f p
⊢ prod
      (Multiset.map (fun (a : B) ↦ HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) a : B[X]))
        (Multiset.filter (Eq b) (Multiset.map (⇑f : (a : A) → B) p.roots))) ∣
    map f p

Tactic:
refine
    (Multiset.prod_dvd_prod_of_le <| Multiset.map_le_map <| Multiset.filter_le (Eq b) _).trans ?_

Premises used: Multiset.prod_dvd_prod_of_le, Multiset.map_le_map, Multiset.filter_le, Eq, Dvd.dvd.trans
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : DecidableEq B
p : A[X]
f : A →+* B
hmap : map f p ≠ 0
b : B
⊢ map f
      (prod
        (Multiset.map (fun (a : A) ↦ HSub.hSub (α := A[X]) (β := A[X]) X ((C : (a : A) → A[X]) a : A[X])) p.roots)) ∣
    map f p

Tactic:
Polynomial.map_dvd f p.prod_multiset_X_sub_C_dvd

Premises used: Polynomial.map_dvd, Polynomial.prod_multiset_X_sub_C_dvd
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : DecidableEq B
p : A[X]
f : A →+* B
hmap : map f p ≠ 0
b : B
h :
  prod
      (Multiset.map (fun (a : B) ↦ HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) a : B[X]))
        (Multiset.map (⇑f : (a : A) → B) p.roots)) =
    map f
      (prod (Multiset.map (fun (a : A) ↦ HSub.hSub (α := A[X]) (β := A[X]) X ((C : (a : A) → A[X]) a : A[X])) p.roots))
⊢ prod
      (Multiset.map (fun (a : B) ↦ HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) a : B[X]))
        (Multiset.map (⇑f : (a : A) → B) p.roots)) ∣
    map f p

Tactic:
convert! Polynomial.map_dvd f p.prod_multiset_X_sub_C_dvd

Premises used: Polynomial.map_dvd, Polynomial.prod_multiset_X_sub_C_dvd
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : DecidableEq B
p : A[X]
f : A →+* B
hmap : map f p ≠ 0
b : B
⊢ prod
      (Multiset.map (fun (a : B) ↦ HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) a : B[X]))
        (Multiset.map (⇑f : (a : A) → B) p.roots)) =
    map f
      (prod (Multiset.map (fun (a : A) ↦ HSub.hSub (α := A[X]) (β := A[X]) X ((C : (a : A) → A[X]) a : A[X])) p.roots))

Tactic:
simp only [Polynomial.map_multiset_prod, Multiset.map_map, Function.comp_apply,
    Polynomial.map_sub, map_X, map_C]

Premises used: Polynomial.map_X, Polynomial.map_sub, Multiset.map_map, Function.comp_apply, Polynomial.map_C, Polynomial.map_multiset_prod
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : DecidableEq B
p : A[X]
f : A →+* B
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
b : B
pos : ∀ (hp0 : p = 0), count b (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ rootMultiplicity b (map f p)
neg : ∀ (hp0 : ¬p = 0), count b (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ rootMultiplicity b (map f p)
⊢ count b (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ rootMultiplicity b (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
inst_2 : IsDomain A
inst_3 : DecidableEq B
p : A[X]
f : A →+* B
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
b : B
hp0 : p = 0
⊢ count b (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ rootMultiplicity b (map f p)

Tactic:
simp only [hp0, roots_zero, Multiset.map_zero, Multiset.count_zero, Polynomial.map_zero,
      rootMultiplicity_zero, le_refl]

Premises used: le_refl, Polynomial.roots_zero, Multiset.map_zero, Multiset.count_zero, 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
inst_2 : IsDomain A
inst_3 : DecidableEq B
p : A[X]
f : A →+* B
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
b : B
hp0 : ¬p = 0
⊢ count b (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ rootMultiplicity b (map f p)

Tactic:
count_map_roots ((Polynomial.map_ne_zero_iff hf).mpr hp0) b

Premises used: Iff.mpr, Polynomial.count_map_roots, 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
inst_2 : IsDomain A
inst_3 : DecidableEq B
p : A[X]
f : A →+* B
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
b : B
hp0 : ¬p = 0
⊢ count b (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ rootMultiplicity b (map f p)

Tactic:
exact count_map_roots ((Polynomial.map_ne_zero_iff hf).mpr hp0) b

Premises used: Iff.mpr, Polynomial.count_map_roots, 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
inst_2 : IsDomain A
inst_3 : IsDomain B
p : A[X]
f : A →+* B
h : map f p ≠ 0
b : B
rw : count b (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ rootMultiplicity b (map f p)
⊢ count b (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ count b (map f p).roots

Tactic:
rw [count_roots]

Premises used: Polynomial.count_roots
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
p : A[X]
f : A →+* B
h : map f p ≠ 0
b : B
⊢ count b (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ rootMultiplicity b (map f p)

Tactic:
apply count_map_roots h

Premises used: Polynomial.count_map_roots
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
p : A[X]
f : A →+* B
h : map f p ≠ 0
⊢ Multiset.map (⇑f : (a : A) → B) p.roots ≤ (map f p).roots

Tactic:
exact Multiset.le_iff_count.2 fun b => by
    rw [count_roots]
    apply count_map_roots h

Premises used: Iff.mpr, Multiset.le_iff_count
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
p : A[X]
f : A →+* B
h : map f p ≠ 0
⊢ Multiset.map (⇑f : (a : A) → B) p.roots ≤ (map f p).roots

Tactic:
classical
  exact Multiset.le_iff_count.2 fun b => by
    rw [count_roots]
    apply count_map_roots h

Premises used: Iff.mpr, Multiset.le_iff_count
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
p : A[X]
f : A →+* B
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
pos : ∀ (hp0 : p = 0), Multiset.map (⇑f : (a : A) → B) p.roots ≤ (map f p).roots
neg : ∀ (hp0 : ¬p = 0), Multiset.map (⇑f : (a : A) → B) p.roots ≤ (map f p).roots
⊢ Multiset.map (⇑f : (a : A) → B) p.roots ≤ (map f p).roots

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
inst_2 : IsDomain A
inst_3 : IsDomain B
p : A[X]
f : A →+* B
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
hp0 : p = 0
⊢ Multiset.map (⇑f : (a : A) → B) p.roots ≤ (map f p).roots

Tactic:
simp only [hp0, roots_zero, Multiset.map_zero, Polynomial.map_zero, le_rfl]

Premises used: le_rfl, Polynomial.roots_zero, Multiset.map_zero, Polynomial.map_zero
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
p : A[X]
f : A →+* B
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
hp0 : ¬p = 0
⊢ Multiset.map (⇑f : (a : A) → B) p.roots ≤ (map f p).roots

Tactic:
map_roots_le ((Polynomial.map_ne_zero_iff hf).mpr hp0)

Premises used: Iff.mpr, Polynomial.map_roots_le, 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
inst_2 : IsDomain A
inst_3 : IsDomain B
p : A[X]
f : A →+* B
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
hp0 : ¬p = 0
⊢ Multiset.map (⇑f : (a : A) → B) p.roots ≤ (map f p).roots

Tactic:
exact map_roots_le ((Polynomial.map_ne_zero_iff hf).mpr hp0)

Premises used: Iff.mpr, Polynomial.map_roots_le, Polynomial.map_ne_zero_iff
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_3
B : Type u_4
inst : Semiring A
inst_1 : CommRing B
inst_2 : IsDomain B
f : A →+* B
p : A[X]
hp0 : p ≠ 0
pos : ∀ (hpm0 : map f p = 0), (↑(map f p).roots.card : WithBot ℕ) ≤ p.degree
neg : ∀ (hpm0 : ¬map f p = 0), (↑(map f p).roots.card : WithBot ℕ) ≤ p.degree
⊢ (↑(map f p).roots.card : WithBot ℕ) ≤ p.degree

Tactic:
by_cases hpm0 : p.map f = 0

Premises used: Polynomial.map
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_3
B : Type u_4
inst : Semiring A
inst_1 : CommRing B
inst_2 : IsDomain B
f : A →+* B
p : A[X]
hp0 : p ≠ 0
hpm0 : map f p = 0
⊢ (↑(map f p).roots.card : WithBot ℕ) ≤ p.degree

Tactic:
simp [hp0, hpm0, zero_le_degree_iff]

Premises used: Polynomial.zero_le_degree_iff
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_3
B : Type u_4
inst : Semiring A
inst_1 : CommRing B
inst_2 : IsDomain B
f : A →+* B
p : A[X]
hp0 : p ≠ 0
hpm0 : ¬map f p = 0
⊢ (↑(map f p).roots.card : WithBot ℕ) ≤ p.degree

Tactic:
card_roots hpm0 |>.trans degree_map_le

Premises used: Polynomial.degree_map_le, Polynomial.card_roots, LE.le.trans
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_3
B : Type u_4
inst : Semiring A
inst_1 : CommRing B
inst_2 : IsDomain B
f : A →+* B
p : A[X]
hp0 : p ≠ 0
hpm0 : ¬map f p = 0
⊢ (↑(map f p).roots.card : WithBot ℕ) ≤ p.degree

Tactic:
exact card_roots hpm0 |>.trans degree_map_le

Premises used: Polynomial.degree_map_le, Polynomial.card_roots, LE.le.trans
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_3
B : Type u_4
inst : Semiring A
inst_1 : CommRing B
inst_2 : IsDomain B
f : A →+* B
p : A[X]
⊢ (map f p).roots.card ≤ p.natDegree

Tactic:
card_roots' _ |>.trans natDegree_map_le

Premises used: Polynomial.card_roots', Polynomial.natDegree_map_le, LE.le.trans
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
inst : CommRing A
p : A[X]
B : Type u_3
inst_1 : CommRing B
inst_2 : IsDomain B
inst_3 : Algebra A B
grw : (p.aroots B).card ≤ p.natDegree
⊢ (p.rootSet B).ncard ≤ p.natDegree

Tactic:
grw [rootSet, Set.ncard_coe_finset, Multiset.toFinset_card_le]

Premises used: Multiset.toFinset_card_le, Polynomial.rootSet, Set.ncard_coe_finset
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
inst : CommRing A
p : A[X]
B : Type u_3
inst_1 : CommRing B
inst_2 : IsDomain B
inst_3 : Algebra A B
⊢ (p.aroots B).card ≤ p.natDegree

Tactic:
p.card_roots_map_le_natDegree

Premises used: Polynomial.card_roots_map_le_natDegree
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
inst : CommRing A
p : A[X]
B : Type u_3
inst_1 : CommRing B
inst_2 : IsDomain B
inst_3 : Algebra A B
⊢ (p.aroots B).card ≤ p.natDegree

Tactic:
exact p.card_roots_map_le_natDegree

Premises used: Polynomial.card_roots_map_le_natDegree
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
inst : CommRing A
p : A[X]
B : Type u_3
inst_1 : CommRing B
inst_2 : IsDomain B
inst_3 : Algebra A B
grw : (↑(p.aroots B).toFinset : Set B).ncard ≤ p.natDegree
grw₁ : #(p.aroots B).toFinset ≤ p.natDegree
grw₂ grw₃ grw₄ : (p.aroots B).card ≤ p.natDegree
⊢ (p.rootSet B).ncard ≤ p.natDegree

Tactic:
classical
  grw [rootSet, Set.ncard_coe_finset, Multiset.toFinset_card_le]
  exact p.card_roots_map_le_natDegree

Premises used: Multiset.toFinset_card_le, Polynomial.rootSet, Polynomial.card_roots_map_le_natDegree, Set.ncard_coe_finset
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
f : A →+* B
inst_4 : DecidablePred fun (x : B) ↦ Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) x
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
p : A[X]
a :
  ∀ (b : B),
    count b
        (Multiset.filter (fun (x : B) ↦ Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) x)
          (map f p).roots) =
      count b (Multiset.map (⇑f : (a : A) → B) p.roots)
⊢ Multiset.filter (fun (x : B) ↦ Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) x)
      (map f p).roots =
    Multiset.map (⇑f : (a : A) → B) p.roots

Tactic:
ext b
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
f : A →+* B
inst_4 : DecidablePred fun (x : B) ↦ Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) x
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
p : A[X]
b : B
a :
  (if Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) b then count b (map f p).roots else 0) =
    count b (Multiset.map (⇑f : (a : A) → B) p.roots)
⊢ count b
      (Multiset.filter (fun (x : B) ↦ Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) x)
        (map f p).roots) =
    count b (Multiset.map (⇑f : (a : A) → B) p.roots)

Tactic:
rw [Multiset.count_filter]

Premises used: Multiset.count_filter
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
f : A →+* B
inst_4 : DecidablePred fun (x : B) ↦ Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) x
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
p : A[X]
b : B
pos :
  ∀ (h : Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) b),
    count b (map f p).roots = count b (Multiset.map (⇑f : (a : A) → B) p.roots)
neg : ∀ (h : b ∉ RingHom.range.{u_1, u_2} (R := A) f), 0 = count b (Multiset.map (⇑f : (a : A) → B) p.roots)
⊢ (if Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) b then count b (map f p).roots else 0) =
    count b (Multiset.map (⇑f : (a : A) → B) p.roots)

Tactic:
split_ifs with h
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
f : A →+* B
inst_4 : DecidablePred fun (x : B) ↦ Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) x
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
p : A[X]
b : B
h : Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) b
pos :
  ∀ (a : A),
    count ((f : (a : A) → B) a) (map f p).roots =
      count.{u_2} (α := B) ((f : (a : A) → B) a : B) (Multiset.map (⇑f : (a : A) → B) p.roots : Multiset B)
⊢ count b (map f p).roots = count b (Multiset.map (⇑f : (a : A) → B) p.roots)

Tactic:
obtain ⟨a, rfl⟩ := h

Premises used: rfl
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
f : A →+* B
inst_4 : DecidablePred fun (x : B) ↦ Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) x
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
p : A[X]
a : A
⊢ count ((f : (a : A) → B) a) (map f p).roots =
    count.{u_2} (α := B) ((f : (a : A) → B) a : B) (Multiset.map (⇑f : (a : A) → B) p.roots : Multiset B)

Tactic:
simp [hf, Multiset.count_map_eq_count', eq_rootMultiplicity_map hf]

Premises used: Polynomial.eq_rootMultiplicity_map, Multiset.count_map_eq_count'
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
f : A →+* B
inst_4 : DecidablePred fun (x : B) ↦ Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) x
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
p : A[X]
b : B
h : b ∉ RingHom.range.{u_1, u_2} (R := A) f
⊢ 0 = count b (Multiset.map (⇑f : (a : A) → B) p.roots)

Tactic:
(Multiset.count_eq_zero.mpr fun h' ↦ h ?_).symm

Premises used: Iff.mpr, Multiset.count_eq_zero, Eq.symm
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
f : A →+* B
inst_4 : DecidablePred fun (x : B) ↦ Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) x
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
p : A[X]
b : B
h : b ∉ RingHom.range.{u_1, u_2} (R := A) f
neg :
  ∀ (h' : Membership.mem (γ := Multiset B) (Multiset.map (⇑f : (a : A) → B) p.roots) b),
    Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) b
⊢ 0 = count b (Multiset.map (⇑f : (a : A) → B) p.roots)

Tactic:
refine (Multiset.count_eq_zero.mpr fun h' ↦ h ?_).symm

Premises used: Iff.mpr, Multiset.count_eq_zero, Eq.symm
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
f : A →+* B
inst_4 : DecidablePred fun (x : B) ↦ Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) x
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
p : A[X]
b : B
h : b ∉ RingHom.range.{u_1, u_2} (R := A) f
h' : Membership.mem (γ := Multiset B) (Multiset.map (⇑f : (a : A) → B) p.roots) b
⊢ Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) b

Tactic:
Exists.imp (fun _ ↦ And.right) <| Multiset.mem_map.mp h'

Premises used: Set.Mem, Multiset.mem_map, setOf, Exists.imp, And.right, Iff.mp, Set.range
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
f : A →+* B
inst_4 : DecidablePred fun (x : B) ↦ Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) x
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
p : A[X]
b : B
h : b ∉ RingHom.range.{u_1, u_2} (R := A) f
h' : Membership.mem (γ := Multiset B) (Multiset.map (⇑f : (a : A) → B) p.roots) b
⊢ Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) b

Tactic:
exact Exists.imp (fun _ ↦ And.right) <| Multiset.mem_map.mp h'

Premises used: Multiset.mem_map, Exists.imp, And.right, Iff.mp
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
f : A →+* B
inst_4 : DecidablePred fun (x : B) ↦ Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) x
hf : Function.Injective.{u_1 + 1, u_2 + 1} (α := A) (β := B) (⇑f : (a : A) → B)
p : A[X]
a :
  ∀ (b : B),
    count b
        (Multiset.filter (fun (x : B) ↦ Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) x)
          (map f p).roots) =
      count b (Multiset.map (⇑f : (a : A) → B) p.roots)
a₁ :
  ∀ (b : B),
    (if Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) b then count b (map f p).roots else 0) =
      count b (Multiset.map (⇑f : (a : A) → B) p.roots)
pos :
  ∀ (b : B) (h : Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) b),
    count b (map f p).roots = count b (Multiset.map (⇑f : (a : A) → B) p.roots)
neg neg₁ :
  ∀ (b : B) (h : b ∉ RingHom.range.{u_1, u_2} (R := A) f), 0 = count b (Multiset.map (⇑f : (a : A) → B) p.roots)
⊢ Multiset.filter (fun (x : B) ↦ Membership.mem (γ := Subring B) (RingHom.range.{u_1, u_2} (R := A) f) x)
      (map f p).roots =
    Multiset.map (⇑f : (a : A) → B) p.roots

Tactic:
classical
  ext b
  rw [Multiset.count_filter]
  split_ifs with h
  · obtain ⟨a, rfl⟩ := h
    simp [hf, Multiset.count_map_eq_count', eq_rootMultiplicity_map hf]
  · refine (Multiset.count_eq_zero.mpr fun h' ↦ h ?_).symm
    exact Exists.imp (fun _ ↦ And.right) <| Multiset.mem_map.mp h'

Premises used: Polynomial.eq_rootMultiplicity_map, Exists.imp, And.right, Multiset.count_filter, Multiset.count_map_eq_count', rfl
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
p : A[X]
f : A →+* B
h : map f p ≠ 0
rw : Multiset.card.{u_2} (α := B) (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ (map f p).roots.card
⊢ p.roots.card ≤ (map f p).roots.card

Tactic:
rw [← p.roots.card_map f]

Premises used: Polynomial.roots, DFunLike.coe, Multiset.card_map
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
p : A[X]
f : A →+* B
h : map f p ≠ 0
⊢ Multiset.card.{u_2} (α := B) (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ (map f p).roots.card

Tactic:
Multiset.card_le_card (map_roots_le h)

Premises used: Multiset.card_le_card, Polynomial.map_roots_le
-- Module: Mathlib.Algebra.Polynomial.Roots
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : IsDomain A
inst_3 : IsDomain B
p : A[X]
f : A →+* B
h : map f p ≠ 0
⊢ Multiset.card.{u_2} (α := B) (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ (map f p).roots.card

Tactic:
exact Multiset.card_le_card (map_roots_le h)

Premises used: Multiset.card_le_card, Polynomial.map_roots_le