awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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
⊢ 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)
have 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
Tactic: rw [map_pow, map_sub, coe_mapRingHom, map_X, map_C]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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
⊢ 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)
have 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
Tactic: rw [map_pow, map_sub, coe_mapRingHom, map_X, map_C]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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
⊢ 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)
have rw₄ : 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
Tactic: rw [map_pow, map_sub, coe_mapRingHom, map_X, map_C]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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
⊢ rootMultiplicity a p = rootMultiplicity ((f : (a : A) → B) a) (map f p)
have pos : ∀ (hp0 : p = 0), rootMultiplicity a p = rootMultiplicity ((f : (a : A) → B) a) (map f p)
Tactic: by_cases hp0 : p = 0
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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)
⊢ rootMultiplicity a p = rootMultiplicity ((f : (a : A) → B) a) (map f p)
have neg : ∀ (hp0 : ¬p = 0), rootMultiplicity a p = rootMultiplicity ((f : (a : A) → B) a) (map f p)
Tactic: by_cases hp0 : p = 0
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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)
have neg : rootMultiplicity ((f : (a : A) → B) a) (map f p) ≤ rootMultiplicity a p
Tactic: apply le_antisymm (le_rootMultiplicity_map ((Polynomial.map_ne_zero_iff hf).mpr hp0) a)
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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 ((f : (a : A) → B) a) (map f p) ≤ rootMultiplicity a p
have 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
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]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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
⊢ count b (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ rootMultiplicity b (map f p)
have 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
Tactic: rw [le_rootMultiplicity_iff hmap, ← Multiset.prod_replicate, ←
    Multiset.map_replicate fun a => X - C a]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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]))
        (replicate (count b (Multiset.map (⇑f : (a : A) → B) p.roots)) b)) ∣
    map f p
have 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
Tactic: rw [← Multiset.filter_eq]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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
have 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
Tactic: refine
    (Multiset.prod_dvd_prod_of_le <| Multiset.map_le_map <| Multiset.filter_le (Eq b) _).trans ?_
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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 p
have 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))
Tactic: convert! Polynomial.map_dvd f p.prod_multiset_X_sub_C_dvd
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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
⊢ count b (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ rootMultiplicity b (map f p)
have pos : ∀ (hp0 : p = 0), count b (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ rootMultiplicity b (map f p)
Tactic: by_cases hp0 : p = 0
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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)
⊢ count b (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ rootMultiplicity b (map f p)
have neg : ∀ (hp0 : ¬p = 0), count b (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ rootMultiplicity b (map f p)
Tactic: by_cases hp0 : p = 0
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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) ≤ count b (map f p).roots
have rw : count b (Multiset.map (⇑f : (a : A) → B) p.roots) ≤ rootMultiplicity b (map f p)
Tactic: rw [count_roots]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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)
⊢ Multiset.map (⇑f : (a : A) → B) p.roots ≤ (map f p).roots
have pos : ∀ (hp0 : p = 0), Multiset.map (⇑f : (a : A) → B) p.roots ≤ (map f p).roots
Tactic: by_cases hp0 : p = 0
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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
⊢ Multiset.map (⇑f : (a : A) → B) p.roots ≤ (map f p).roots
have neg : ∀ (hp0 : ¬p = 0), Multiset.map (⇑f : (a : A) → B) p.roots ≤ (map f p).roots
Tactic: by_cases hp0 : p = 0
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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
⊢ (↑(map f p).roots.card : WithBot ℕ) ≤ p.degree
have pos : ∀ (hpm0 : map f p = 0), (↑(map f p).roots.card : WithBot ℕ) ≤ p.degree
Tactic: by_cases hpm0 : p.map f = 0
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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
⊢ (↑(map f p).roots.card : WithBot ℕ) ≤ p.degree
have neg : ∀ (hpm0 : ¬map f p = 0), (↑(map f p).roots.card : WithBot ℕ) ≤ p.degree
Tactic: by_cases hpm0 : p.map f = 0
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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.rootSet B).ncard ≤ p.natDegree
have grw : (p.aroots B).card ≤ p.natDegree
Tactic: grw [rootSet, Set.ncard_coe_finset, Multiset.toFinset_card_le]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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.rootSet B).ncard ≤ p.natDegree
have grw : (↑(p.aroots B).toFinset : Set B).ncard ≤ p.natDegree
Tactic: classical
  grw [rootSet, Set.ncard_coe_finset, Multiset.toFinset_card_le]
  exact p.card_roots_map_le_natDegree
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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
⊢ (p.rootSet B).ncard ≤ p.natDegree
have grw₁ : #(p.aroots B).toFinset ≤ p.natDegree
Tactic: classical
  grw [rootSet, Set.ncard_coe_finset, Multiset.toFinset_card_le]
  exact p.card_roots_map_le_natDegree
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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
⊢ (p.rootSet B).ncard ≤ p.natDegree
have grw₂ : (p.aroots B).card ≤ p.natDegree
Tactic: classical
  grw [rootSet, Set.ncard_coe_finset, Multiset.toFinset_card_le]
  exact p.card_roots_map_le_natDegree
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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]
⊢ 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
have 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)
Tactic: ext b
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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
⊢ 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)
have 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)
Tactic: rw [Multiset.count_filter]
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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
⊢ (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)
have 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)
Tactic: split_ifs with h
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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)
⊢ (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)
have neg : ∀ (h : b ∉ RingHom.range.{u_1, u_2} (R := A) f), 0 = count b (Multiset.map (⇑f : (a : A) → B) p.roots)
Tactic: split_ifs with h
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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
⊢ count b (map f p).roots = count b (Multiset.map (⇑f : (a : A) → B) p.roots)
have 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)
Tactic: obtain ⟨a, rfl⟩ := h
Cleared hypotheses: b, h
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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)
have 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
Tactic: refine (Multiset.count_eq_zero.mpr fun h' ↦ h ?_).symm
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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]
⊢ 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
have 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)
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'
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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
have 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)
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'
-- Lean 4 proof step from Mathlib.Algebra.Polynomial.Roots (goal)
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)
⊢ 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
have 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)
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'