mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : CommRing R
p : R[X]
t : R
h : p ≠ 0
x✝ : p.IsRoot t ∧ (derivative p).IsRoot t
h0 : p.IsRoot t
h1 : (derivative p).IsRoot t
m : ℕ
hm : m + 1 + 1 ≤ 1
⊢ ((⇑derivative)^[m + 1 + 1] p).IsRoot t

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : CommRing R
p : R[X]
t : R
h : p ≠ 0
⊢ 1 < rootMultiplicity t p ↔ p.IsRoot t ∧ (derivative p).IsRoot t

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : GCDMonoid R[X]
p : R[X]
t : R
h : p ≠ 0
⊢ 1 < rootMultiplicity t p ↔ (gcd p (derivative p)).IsRoot t

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : CharZero R
p : R[X]
t : R
hpt : p.IsRoot t
⊢ rootMultiplicity t (derivative p) = rootMultiplicity t p - 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : CharZero R
p : R[X]
t : R
⊢ rootMultiplicity t p - 1 ≤ rootMultiplicity t (derivative p)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : CharZero R
p : R[X]
t : R
n : ℕ
h : p ≠ 0
hroot : ∀ m ≤ n, ((⇑derivative)^[m] p).IsRoot t
⊢ n ! ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : CharZero R
f g : R[X]
hf0 : f ≠ 0
a : R
haf : f.IsRoot a
r : R[X]
hr : derivative f * g = f * r
⊢ derivative f ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : IsCancelMulZero α
inst✝ : Nontrivial α
p : FactorSet α
⊢ p.prod = 0 ↔ p = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : CharZero R
f g : R[X]
hf0 : f ≠ 0
a : R
haf : f.IsRoot a
r : R[X]
hr : derivative f * g = f * r
hdf0 : derivative f ≠ 0
hg : ¬g.IsRoot a
⊢ g ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
p : Associates α
hp : Irreducible p
s : Multiset { a // Irreducible a }
⊢ p.count ↑s = Multiset.count ⟨p, hp⟩ s

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
p : Associates α
hp : Irreducible p
⊢ p.count 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝ : CommMonoidWithZero α
p : Associates α
hp : Irreducible p
⊢ p ∈ ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : CharZero R
f g : R[X]
hf0 : f ≠ 0
hfd : f ∣ derivative f * g
a : R
haf : f.IsRoot a
⊢ g.IsRoot a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : NormalizationMonoid R
p : R[X]
⊢ C ↑(normUnit p.leadingCoeff) * C ↑(normUnit p.leadingCoeff)⁻¹ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : NormalizationMonoid R
p : R[X]
⊢ C ↑(normUnit p.leadingCoeff)⁻¹ * C ↑(normUnit p.leadingCoeff) = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝ : CommMonoidWithZero α
p : Associates α
hp : Irreducible p
l : Multiset { a // Irreducible a }
⊢ p ∈ ↑l ↔ ⟨p, hp⟩ ∈ l

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : NormalizationMonoid R
⊢ ↑{ val := C ↑(normUnit (leadingCoeff 0)), inv := C ↑(normUnit (leadingCoeff 0))⁻¹, val_inv := ⋯, inv_val := ⋯ } = ↑1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : NormalizationMonoid R
⊢ ↑{ val := C ↑(normUnit (leadingCoeff 1)), inv := C ↑(normUnit (leadingCoeff 1))⁻¹, val_inv := ⋯, inv_val := ⋯ } = ↑1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝ : CommMonoidWithZero α
p : Associates α
hp : ¬Irreducible p
s : FactorSet α
h : p ∈ s
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
⊢ ∃ q, (Multiset.map (fun a ↦ X - C a) p.roots).prod * q = p ∧ p.roots.card + q.natDegree = p.natDegree ∧ q.roots = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : NormalizationMonoid R
a✝ : R[X]
u : R[X]ˣ
h : a✝ ≠ 0
⊢ ↑{ val := C ↑(normUnit (a✝ * ↑u).leadingCoeff), inv := C ↑(normUnit (a✝ * ↑u).leadingCoeff)⁻¹, val_inv := ⋯,
        inv_val := ⋯ } =
    ↑(u⁻¹ *
        { val := C ↑(normUnit a✝.leadingCoeff), inv := C ↑(normUnit a✝.leadingCoeff)⁻¹, val_inv := ⋯, inv_val := ⋯ })

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n : ℕ
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : StrongNormalizationMonoid R
a✝ b✝ : R[X]
hp0 : a✝ ≠ 0
hq0 : b✝ ≠ 0
⊢ ↑(normUnit (a✝ * b✝)) = ↑(normUnit a✝ * normUnit b✝)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : NormalizationMonoid R
p : R[X]
⊢ ↑(normUnit p) = C ↑(normUnit p.leadingCoeff)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : NormalizationMonoid R
p : R[X]
⊢ (normalize p).leadingCoeff = normalize p.leadingCoeff

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : NormalizationMonoid R
p : R[X]
hp : p.Monic
⊢ normalize p = p

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hp : p.Monic
hroots : p.roots.card = p.natDegree
⊢ (Multiset.map (fun a ↦ X - C a) p.roots).prod = p

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizationMonoid R
p : R[X]
⊢ (normalize p).roots = p.roots

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : NormalizationMonoid R
⊢ normUnit X = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
a p : R[X]
hp : p.Monic
hap : a ∣ p
⊢ a.leadingCoeff ∣ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : NoZeroDivisors R
inst✝ : NormalizationMonoid R
⊢ X = normalize X

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : DivisionRing R
p : R[X]
hp0 : C (p.coeff 0) ≠ 0
hp : ¬IsUnit (C (p.coeff 0))
h : p.degree ≤ 0
⊢ ¬C (p.coeff 0) = C 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : DivisionRing R
p : R[X]
hp0 : p ≠ 0
hp : ¬IsUnit p
h : p.degree ≤ 0
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : Field R
p : R[X]
h : p.degree = 0
⊢ p.degree ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hirr : Irreducible p
⊢ p.roots.card ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : Field R
p : R[X]
h : p.degree = 0
this : p.degree ≤ 0
hc : p.coeff 0 = 0
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : FactorSet α
h : p.prod = q.prod
⊢ p = q

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hirr : Irreducible p
hdeg : p.natDegree ≠ 1
⊢ p.roots = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a : α
⊢ Multiset.map Subtype.val (factors' a) = Multiset.map Associates.mk (factors a)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : Field R
p : R[X]
h : p.degree = 0
this : p.degree ≤ 0
hc : p.coeff 0 ≠ 0
⊢ 1 = p * C (p.coeff 0)⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : α
h : a ~ᵤ b
ha : a = 0
⊢ b = 0

Tactic:
← associated_zero_iff_eq_zero,

Goal state after:
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : α
h : a ~ᵤ b
ha : a = 0
⊢ b ~ᵤ 0
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : α
h : a ~ᵤ b
hb : b ≠ 0
⊢ a ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : α
h : a ~ᵤ b
⊢ factors' a = factors' b

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Content

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
p✝ q✝ : R[X]
a✝ : Nontrivial R
n : ℕ
ih : ∀ (p q : R[X]), (p * q).degree < ↑n → Associated (p * q).content (p.content * q.content)
p q : R[X]
p0 : ¬p = 0
q0 : ¬q = 0
heq : p.primPart.degree + q.primPart.degree = ↑n
h : (q.primPart * p.primPart).content = 1
⊢ Associated (C p.content * p.primPart * (C q.content * q.primPart)).content
    ((C p.content * p.primPart).content * (C q.content * q.primPart).content)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : Field R
p q : R[X]
⊢ q * p.div q + p.mod q = p

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : Field R
q p : R[X]
hq : q ≠ 0
⊢ (p.mod q).degree < q.degree

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a : Associates α
⊢ FactorSet α

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a : α
h : a ≠ 0
⊢ (Associates.mk a).factors = ↑(factors' a)

Tactic:
apply dif_neg

Goal state after:
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a : α
h : a ≠ 0
⊢ ¬Associates.mk a = 0
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a : α
h : a ≠ 0
⊢ (Associates.mk a).factors = ↑(factors' a)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n : ℕ
inst✝ : Field R
p q : R[X]
⊢ ∀ (a : R[X]), a / 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a : Associates α
⊢ a.factors.prod = a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : Field R
p q : R[X]
hq0 : q ≠ 0
h : p.degree < q.degree
⊢ p.degree < (q * C q.leadingCoeff⁻¹).degree

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Subsingleton α
a : Associates α
⊢ a.factors = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a : Associates α
a✝ : Nontrivial α
h : a.factors = ⊤
⊢ a = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a : Associates α
⊢ a.factors = ⊤ ↔ a = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a : Associates α
⊢ (∃ s, a.factors = ↑s) ↔ a ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : Associates α
h : a.factors = b.factors
⊢ a.factors.prod = b.factors.prod

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : Field R
p q : R[X]
hq0 : q ≠ 0
h : p.degree < q.degree
⊢ p % q = p

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
p_monic : p.Monic
p_1 : p ≠ 1
⊢ Irreducible p ↔ ∀ (q : R[X]), q.degree ≤ ↑(p.natDegree / 2) → q ∣ p → IsUnit q

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : Field R
p q : R[X]
hq0 : q ≠ 0
h : p / q = 0
⊢ p.degree < q.degree

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : Associates α
h : a.factors = b.factors
⊢ a = b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : Field R
p q : R[X]
hq0 : q ≠ 0
h : p.degree < q.degree
⊢ p / q = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : Associates α
⊢ (a * b).factors = a.factors + b.factors

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
p : A[X]
f : A →+* B
hmap : map f p ≠ 0
a : A
⊢ rootMultiplicity a p ≤ rootMultiplicity (f a) (map f p)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : Field R
p q : R[X]
hq0 : q ≠ 0
hpq : q.degree ≤ p.degree
⊢ q.degree + (p / q).degree = p.degree

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
s t : Associates α
d : Associates α
eq : t = s * d
⊢ s.factors ≤ t.factors

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : Associates α
⊢ a.factors ≤ b.factors ↔ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : Field R
p q : R[X]
⊢ (p / q).degree ≤ p.degree

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : Field R
p q : R[X]
hp : p ≠ 0
hq : 0 < q.degree
hq0 : q = 0
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : Field R
p q : R[X]
hp : p ≠ 0
hq : 0 < q.degree
hq0 : q ≠ 0
⊢ 0 < (q * C q.leadingCoeff⁻¹).degree

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹ : CommRing A
inst✝ : CommRing B
p : A[X]
f : A →+* B
hf : Function.Injective ⇑f
a : A
⊢ rootMultiplicity a p = rootMultiplicity (f a) (map f p)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : Field R
p q : R[X]
hp : p ≠ 0
hq : 0 < q.degree
⊢ (p / q).degree < p.degree

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
k : Type y
inst✝¹ : Field R
p : R[X]
inst✝ : Field k
f : R →+* k
⊢ IsUnit (map f p) ↔ IsUnit p

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝³ : CommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
h : ∀ (p : Associates α), Irreducible p → p.count a.factors = p.count b.factors
⊢ a.factors = b.factors

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : IsDomain A
inst✝ : DecidableEq B
p : A[X]
f : A →+* B
hmap : map f p ≠ 0
b : B
⊢ count b (Multiset.map (⇑f) p.roots) ≤ rootMultiplicity b (map f p)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
k : Type y
inst✝¹ : Field R
p q : R[X]
inst✝ : Field k
f : R →+* k
⊢ map f (p / q) = map f p / map f q

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝³ : CommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b p : Associates α
hb : b ≠ 0
hp : Irreducible p
h : a.factors ≤ b.factors
⊢ p.count a.factors ≤ p.count b.factors

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : IsDomain A
inst✝ : DecidableEq B
p : A[X]
f : A →+* B
hf : Function.Injective ⇑f
b : B
⊢ count b (Multiset.map (⇑f) p.roots) ≤ rootMultiplicity b (map f p)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
a b : FactorSet α
⊢ a.prod ≤ b.prod ↔ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : IsDomain A
inst✝ : IsDomain B
p : A[X]
f : A →+* B
h : map f p ≠ 0
b : B
⊢ count b (Multiset.map (⇑f) p.roots) ≤ count b (map f p).roots

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : IsDomain A
inst✝ : IsDomain B
p : A[X]
f : A →+* B
h : map f p ≠ 0
⊢ Multiset.map (⇑f) p.roots ≤ (map f p).roots

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
k : Type y
inst✝¹ : Field R
p q : R[X]
inst✝ : Field k
f : R →+* k
⊢ map f (p % q) = map f p % map f q

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : IsDomain A
inst✝ : IsDomain B
p : A[X]
f : A →+* B
hf : Function.Injective ⇑f
⊢ Multiset.map (⇑f) p.roots ≤ (map f p).roots

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: k : Type y
inst✝ : Field k
p q : k[X]
hq : q.natDegree ≠ 0
⊢ q.leadingCoeff ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a p : Associates α
hm : p ∈ a.factors
⊢ p ∣ a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Content

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
⊢ Associated (p * q).content (p.content * q.content)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Content

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
inst✝ : StrongNormalizedGCDMonoid R
p q : R[X]
⊢ normalize (p.content * q.content) = p.content * q.content

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a : α
p : Associates α
hp : Irreducible p
hz : a ≠ 0
h_mem : ⟨p, hp⟩ ∈ factors' a
⊢ p ∣ Associates.mk a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: A : Type u_3
B : Type u_4
inst✝² : Semiring A
inst✝¹ : CommRing B
inst✝ : IsDomain B
f : A →+* B
p : A[X]
hp0 : p ≠ 0
⊢ ↑(map f p).roots.card ≤ p.degree

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Content

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
⊢ (p * q).IsPrimitive

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
⊢ ⟨Associates.mk p, ⋯⟩ ∈ factors' a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: A : Type u_1
inst✝³ : CommRing A
p : A[X]
B : Type u_3
inst✝² : CommRing B
inst✝¹ : IsDomain B
inst✝ : Algebra A B
⊢ (p.rootSet B).ncard ≤ p.natDegree

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ ⟨Associates.mk p, ⋯⟩ ∈ factors' a ↔ p ∣ a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
⊢ Associates.mk p ∈ (Associates.mk a).factors

Tactic:
by

Goal state after: