mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝ : CommMonoidWithZero α
p : Associates α
hp : Irreducible p
⊢ if hp : Irreducible p then BfactorSetMem ⟨p, hp⟩ ⊤ else False

Tactic:
;

Goal state after:
α : Type u_1
inst✝ : CommMonoidWithZero α
p : Associates α
hp : Irreducible p
⊢ if hp : Irreducible p then BfactorSetMem ⟨p, hp⟩ ⊤ else False
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: α : Type u_1
inst✝ : CommMonoidWithZero α
p : Associates α
hp : Irreducible p
⊢ BfactorSetMem ⟨p, hp⟩ ⊤

Tactic:
;

Goal state after:
α : Type u_1
inst✝ : CommMonoidWithZero α
p : Associates α
hp : Irreducible p
⊢ BfactorSetMem ⟨p, hp⟩ ⊤
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 }
⊢ FactorSetMem (↑l) p ↔ l.Mem ⟨p, hp⟩

Tactic:
;

Goal state after:
α : Type u_1
inst✝ : CommMonoidWithZero α
p : Associates α
hp : Irreducible p
l : Multiset { a // Irreducible a }
⊢ FactorSetMem (↑l) p ↔ l.Mem ⟨p, hp⟩
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 }
⊢ (if hp : Irreducible p then BfactorSetMem ⟨p, hp⟩ ↑l else False) ↔ l.Mem ⟨p, hp⟩

Tactic:
;

Goal state after:
α : Type u_1
inst✝ : CommMonoidWithZero α
p : Associates α
hp : Irreducible p
l : Multiset { a // Irreducible a }
⊢ (if hp : Irreducible p then BfactorSetMem ⟨p, hp⟩ ↑l else False) ↔ l.Mem ⟨p, hp⟩
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 }
⊢ BfactorSetMem ⟨p, hp⟩ ↑l ↔ l.Mem ⟨p, hp⟩

Tactic:
;

Goal state after:
α : Type u_1
inst✝ : CommMonoidWithZero α
p : Associates α
hp : Irreducible p
l : Multiset { a // Irreducible a }
⊢ BfactorSetMem ⟨p, hp⟩ ↑l ↔ l.Mem ⟨p, hp⟩
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✝¹ a✝ : Multiset { a // Irreducible a }
h : prod ↑a✝¹ = prod ↑a✝
a' : Associates α
irred : Irreducible a'
ha : ↑⟨a', irred⟩ ∈ Multiset.map Subtype.val a✝¹
⊢ Irreducible ↑⟨a', irred⟩

Tactic:
rwa [Subtype.coe_mk]

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✝¹ a✝ : Multiset { a // Irreducible a }
h : prod ↑a✝¹ = prod ↑a✝
a' : Associates α
irred : Irreducible a'
ha : ↑⟨a', irred⟩ ∈ Multiset.map Subtype.val a✝
⊢ Irreducible ↑⟨a', irred⟩

Tactic:
rwa [Subtype.coe_mk]

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

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

Tactic:
;

Goal state after:
R : Type u
inst✝ : Field R
p : R[X]
h : (C 0).degree = 0
this : p.degree ≤ 0
hc : p.coeff 0 = 0
⊢ False
Module: Mathlib.Algebra.Polynomial.FieldDivision

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

Tactic:
;

Goal state after:
R : Type u
inst✝ : Field R
p : R[X]
this : p.degree ≤ 0
hc : p.coeff 0 = 0
h : degree 0 = 0
⊢ False
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
this : a.factors.prod = b.factors.prod
⊢ a = b

Tactic:
factors_prod,

Goal state after:
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : Associates α
h : a.factors = b.factors
this : a = b.factors.prod
⊢ a = b
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✝ : Nontrivial α
⊢ (a * b).factors.prod.factors = (a.factors.prod * b.factors.prod).factors

Tactic:
factors_prod,

Goal state after:
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : Associates α
a✝ : Nontrivial α
⊢ (a * b).factors = (a.factors.prod * b.factors.prod).factors
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✝ : Nontrivial α
⊢ (a * b).factors = (a.factors.prod * b.factors.prod).factors

Tactic:
factors_prod,

Goal state after:
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : Associates α
a✝ : Nontrivial α
⊢ (a * b).factors = (a * b.factors.prod).factors
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: case pos
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
hp0 : p = 0
⊢ rootMultiplicity a p = rootMultiplicity (f a) (map f p)

pp: case neg
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
hp0 : ¬p = 0
⊢ rootMultiplicity a p = rootMultiplicity (f a) (map f p)

Tactic:
;

Goal state after:
case pos
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
hp0 : p = 0
⊢ rootMultiplicity a p = rootMultiplicity (f a) (map f p)

case neg
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
hp0 : ¬p = 0
⊢ rootMultiplicity a p = rootMultiplicity (f a) (map f p)
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 ≤ s.factors + d.factors

Tactic:
;

Goal state after:
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
s t : Associates α
d : Associates α
eq : t = s * d
⊢ s.factors ≤ s.factors + d.factors
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
this : a.factors.prod ≤ b.factors.prod
⊢ a ≤ b

Tactic:
factors_prod,

Goal state after:
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : Associates α
h : a.factors ≤ b.factors
this : a ≤ b.factors.prod
⊢ a ≤ b
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: case neg
R : Type u
inst✝ : Field R
p q : R[X]
hq : ¬q = 0
⊢ (p /ₘ (q * C q.leadingCoeff⁻¹)).degree ≤ p.degree

Tactic:
;

Goal state after:
case neg
R : Type u
inst✝ : Field R
p q : R[X]
hq : ¬q = 0
⊢ (p /ₘ (q * C q.leadingCoeff⁻¹)).degree ≤ p.degree
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.degree

Tactic:
;

Goal state after:
R : Type u
inst✝ : Field R
p q : R[X]
hp : p ≠ 0
hq : 0 < q.degree
hq0 : q ≠ 0
⊢ 0 < q.degree
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: case neg
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
ha : ¬a = 0
sa : Multiset { p // Irreducible p }
h_sa : a.factors = ↑sa
sb : Multiset { p // Irreducible p }
h : ↑sa ≤ ↑sb
h_sb : b.factors = ↑sb
⊢ Multiset.count ⟨p, hp⟩ sa ≤ Multiset.count ⟨p, hp⟩ sb

Tactic:
;

Goal state after:
case neg
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
ha : ¬a = 0
sa : Multiset { p // Irreducible p }
h_sa : a.factors = ↑sa
sb : Multiset { p // Irreducible p }
h : ↑sa ≤ ↑sb
h_sb : b.factors = ↑sb
⊢ Multiset.count ⟨p, hp⟩ sa ≤ Multiset.count ⟨p, hp⟩ sb
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
⊢ q.primPart.eraseLead.degree < q.primPart.degree

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
⊢ p.primPart.degree ≠ ⊥

Tactic:
· apply degree_erase_lt q.primPart_ne_zero

Goal state after:
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
⊢ p.primPart.degree ≠ ⊥
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: case inr
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a p : Associates α
ha0 : a ≠ 0
a0 : α
hm : p ∈ ↑(factors' a0)
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
⊢ p ∈ Multiset.map Subtype.val (factors' a0)

Tactic:
;

Goal state after:
case inr
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a p : Associates α
ha0 : a ≠ 0
a0 : α
hm : p ∈ ↑(factors' a0)
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
⊢ p ∈ Multiset.map Subtype.val (factors' a0)
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
q : α
hq : q ∈ UniqueFactorizationMonoid.factors a
hpq : p ~ᵤ q
⊢ ∃ a_1, ∃ (h : a_1 ∈ UniqueFactorizationMonoid.factors a), ⟨Associates.mk a_1, ⋯⟩ = ⟨Associates.mk p, ⋯⟩

Tactic:
;

Goal state after:
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
q : α
hq : q ∈ UniqueFactorizationMonoid.factors a
hpq : p ~ᵤ q
⊢ ∃ a_1, ∃ (h : a_1 ∈ UniqueFactorizationMonoid.factors a), ⟨Associates.mk a_1, ⋯⟩ = ⟨Associates.mk p, ⋯⟩
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: case h
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
q : α
hq : q ∈ UniqueFactorizationMonoid.factors a
hpq : p ~ᵤ q
⊢ ∃ (h : q ∈ UniqueFactorizationMonoid.factors a), ⟨Associates.mk q, ⋯⟩ = ⟨Associates.mk p, ⋯⟩

Tactic:
;

Goal state after:
case h
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
q : α
hq : q ∈ UniqueFactorizationMonoid.factors a
hpq : p ~ᵤ q
⊢ ∃ (h : q ∈ UniqueFactorizationMonoid.factors a), ⟨Associates.mk q, ⋯⟩ = ⟨Associates.mk p, ⋯⟩
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : Field R
p₁ p₂ q : R[X]
⊢ (p₁ - p₂) % q = p₁ % q - p₂ % q

Tactic:
by
  simp [Polynomial.mod_def, Polynomial.sub_modByMonic]

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

Goal state before:
pp: R : Type u
k : Type y
inst✝³ : Field R
p q : R[X]
inst✝² : Field k
inst✝¹ : DecidableEq R
inst✝ : DecidableEq k
f : R →+* k
x : R[X]
⊢ EuclideanDomain.gcd 0 (map f x) = map f (EuclideanDomain.gcd 0 x)

Tactic:
EuclideanDomain.gcd_zero_left

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

Goal state before:
pp: R : Type u
k : Type y
inst✝² : Field R
inst✝¹ : CommSemiring k
inst✝ : DecidableEq R
ϕ : R →+* k
f g : R[X]
α : k
hf : eval₂ ϕ α f = 0
hg : eval₂ ϕ α g = 0
⊢ eval₂ ϕ α (EuclideanDomain.gcd f g) = 0

Tactic:
EuclideanDomain.gcd_eq_gcd_ab f g,

Goal state after:
R : Type u
k : Type y
inst✝² : Field R
inst✝¹ : CommSemiring k
inst✝ : DecidableEq R
ϕ : R →+* k
f g : R[X]
α : k
hf : eval₂ ϕ α f = 0
hg : eval₂ ϕ α g = 0
⊢ eval₂ ϕ α (f * EuclideanDomain.gcdA f g + g * EuclideanDomain.gcdB f g) = 0
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 : Associates α
hp : Irreducible p
⊢ p.factors.prod = FactorSet.prod ↑{⟨p, hp⟩}

Tactic:
factors_prod,

Goal state after:
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p : Associates α
hp : Irreducible p
⊢ p = FactorSet.prod ↑{⟨p, hp⟩}
Module: Mathlib.RingTheory.Polynomial.UniqueFactorization

Goal state before:
pp: σ : Type v
D : Type u
inst✝¹ : CommRing D
inst✝ : UniqueFactorizationMonoid D
⊢ UniqueFactorizationMonoid D[X]

Tactic:
by
  let := Classical.arbitrary (NormalizedGCDMonoid D)
  exact ufm_of_decomposition_of_wfDvdMonoid

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 : Associates α
hp : Irreducible p
⊢ p =
    match ↑{⟨p, hp⟩} with
    | none => 0
    | some s => (Multiset.map Subtype.val s).prod

Tactic:
;

Goal state after:
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p : Associates α
hp : Irreducible p
⊢ p =
    match ↑{⟨p, hp⟩} with
    | none => 0
    | some s => (Multiset.map Subtype.val s).prod
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 : Associates α
hp : Irreducible p
⊢ p = {p}.prod

Tactic:
;

Goal state after:
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p : Associates α
hp : Irreducible p
⊢ p = {p}.prod
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
h✝ : ∃ n r, r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r
r : R[X]
rdeg : r.natDegree = Nat.find h✝
rprim : r.IsPrimitive
pr : p ∣ r
qr : q ∣ r
con : ∃ n s, s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : s.natDegree = Nat.find con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : r.natDegree ≤ s.natDegree
sC : 0 < s.natDegree
hcancel : (r.cancelLeads s).natDegree < Nat.find con
rcs : r ∣ C r.leadingCoeff * s - C s.leadingCoeff * X ^ (s.natDegree - r.natDegree) * r
h : r ∣ C r.leadingCoeff * s
a✝ : Nontrivial R
hC0 : ¬C r.leadingCoeff = 0
⊢ r ∣ s.primPart

Tactic:
← rprim.dvd_primPart_iff_dvd (mul_ne_zero hC0 s0)

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
h✝ : ∃ n r, r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r
r : R[X]
rdeg : r.natDegree = Nat.find h✝
rprim : r.IsPrimitive
pr : p ∣ r
qr : q ∣ r
con : ∃ n s, s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : s.natDegree = Nat.find con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : r.natDegree ≤ s.natDegree
sC : 0 < s.natDegree
hcancel : (r.cancelLeads s).natDegree < Nat.find con
rcs : r ∣ C r.leadingCoeff * s - C s.leadingCoeff * X ^ (s.natDegree - r.natDegree) * r
h : r ∣ (C r.leadingCoeff * s).primPart
a✝ : Nontrivial R
hC0 : ¬C r.leadingCoeff = 0
⊢ r ∣ s.primPart
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 : Associates α
hp : Irreducible p
k : ℕ
⊢ p ^ k = (Multiset.map Subtype.val (replicate k ⟨p, hp⟩)).prod

Tactic:
;

Goal state after:
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p : Associates α
hp : Irreducible p
k : ℕ
⊢ p ^ k = (Multiset.map Subtype.val (replicate k ⟨p, hp⟩)).prod
Module: Mathlib.RingTheory.Polynomial.UniqueFactorization

Goal state before:
pp: σ : Type v
D : Type u
inst✝¹ : CommRing D
inst✝ : UniqueFactorizationMonoid D
d : ℕ
a : MvPolynomial σ D
ha : a ≠ 0
⊢ ∃ f, (∀ b ∈ f, Prime b) ∧ Associated f.prod a

Tactic:
;

Goal state after:
σ : Type v
D : Type u
inst✝¹ : CommRing D
inst✝ : UniqueFactorizationMonoid D
d : ℕ
a : MvPolynomial σ D
ha : a ≠ 0
⊢ ∃ f, (∀ b ∈ f, Prime b) ∧ Associated f.prod a
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: case zero
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
a : Associates α
⊢ factors 1 = 0

Tactic:
;

Goal state after:
case zero
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
a : Associates α
⊢ factors 1 = 0
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: case neg.inl
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
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hz : p.count a.factors = 0
⊢ p.count a.factors = 0 ∨ p.count a.factors = p.count (a * b).factors

pp: case neg.inr
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
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb0 : p.count b.factors = 0
⊢ p.count a.factors = 0 ∨ p.count a.factors = p.count (a * b).factors

Tactic:
;

Goal state after:
case neg.inl
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
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hz : p.count a.factors = 0
⊢ p.count a.factors = 0 ∨ p.count a.factors = p.count (a * b).factors

case neg.inr
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
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb0 : p.count b.factors = 0
⊢ p.count a.factors = 0 ∨ p.count a.factors = p.count (a * b).factors
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: case neg
R : Type u
inst✝¹ : Field R
p q : R[X]
inst✝ : DecidableEq R
hq : q ≠ 0
hp : ¬p = 0
⊢ p ∈ normalizedFactors q ↔ Irreducible p ∧ p.Monic ∧ p ∣ q

Tactic:
mem_normalizedFactors_iff' hq,

Goal state after:
case neg
R : Type u
inst✝¹ : Field R
p q : R[X]
inst✝ : DecidableEq R
hq : q ≠ 0
hp : ¬p = 0
⊢ Irreducible p ∧ normalize p = p ∧ p ∣ q ↔ Irreducible p ∧ p.Monic ∧ p ∣ q
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: case pos
F : Type u_1
inst✝ : DivisionSemiring F
f : F[X]
H : f.natDegree = 0
x : F
h : Irreducible 0
hf : C x = f
hx : x = 0
⊢ False

Tactic:
;

Goal state after:
case pos
F : Type u_1
inst✝ : DivisionSemiring F
f : F[X]
H : f.natDegree = 0
x : F
h : Irreducible 0
hf : C x = f
hx : x = 0
⊢ False
Module: Mathlib.Algebra.AlgebraicCard

Goal state before:
pp: R : Type u
A : Type v
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : CommRing A
inst✝² : IsDomain A
inst✝¹ : Algebra R A
inst✝ : Module.IsTorsionFree R A
⊢ lift.{v, u} (max #R ℵ₀) * ℵ₀ ≤ max (lift.{v, u} #R) ℵ₀

Tactic:
;

Goal state after:
R : Type u
A : Type v
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : CommRing A
inst✝² : IsDomain A
inst✝¹ : Algebra R A
inst✝ : Module.IsTorsionFree R A
⊢ lift.{v, u} (max #R ℵ₀) * ℵ₀ ≤ max (lift.{v, u} #R) ℵ₀
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: case succ
α : Type u_1
inst✝⁴ : CommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : DecidableEq (Associates α)
inst✝¹ : (p : Associates α) → Decidable (Irreducible p)
inst✝ : Nontrivial α
a : Associates α
ha : a ≠ 0
p : Associates α
hp : Irreducible p
n : ℕ
h : p.count (a ^ n).factors = n * p.count a.factors
⊢ p.count a.factors + n * p.count a.factors = (n + 1) * p.count a.factors

Tactic:
;

Goal state after:
case succ
α : Type u_1
inst✝⁴ : CommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : DecidableEq (Associates α)
inst✝¹ : (p : Associates α) → Decidable (Irreducible p)
inst✝ : Nontrivial α
a : Associates α
ha : a ≠ 0
p : Associates α
hp : Irreducible p
n : ℕ
h : p.count (a ^ n).factors = n * p.count a.factors
⊢ p.count a.factors + n * p.count a.factors = (n + 1) * p.count a.factors
Module: Mathlib.RingTheory.Algebraic.Basic

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
h : ¬Function.Injective ⇑(algebraMap R A)
a : A
inj : Function.Injective ⇑(aeval a)
⊢ ⇑(algebraMap R A) = ⇑(aeval a) ∘ ⇑C

Tactic:
;

Goal state after:
R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
h : ¬Function.Injective ⇑(algebraMap R A)
a : A
inj : Function.Injective ⇑(aeval a)
⊢ ⇑(algebraMap R A) = ⇑(aeval a) ∘ ⇑C
Module: Mathlib.RingTheory.Algebraic.Basic

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
h : ¬Function.Injective ⇑(algebraMap R A)
a : A
inj : Function.Injective ⇑(aeval a)
x✝ : R
⊢ (algebraMap R A) x✝ = (⇑(aeval a) ∘ ⇑C) x✝

Tactic:
;

Goal state after:
R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
h : ¬Function.Injective ⇑(algebraMap R A)
a : A
inj : Function.Injective ⇑(aeval a)
x✝ : R
⊢ (algebraMap R A) x✝ = (⇑(aeval a) ∘ ⇑C) x✝
Module: Mathlib.Algebra.Central.End

Goal state before:
pp: R : Type u_1
M : Type u_3
inst✝¹⁶ : Semiring R
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : Module R M
inst✝¹³ : Free R M
S : Type u_4
M₂ : Type u_5
inst✝¹² : CommRing S
inst✝¹¹ : IsCancelMulZero S
inst✝¹⁰ : Module S M
inst✝⁹ : SMulCommClass R S M
inst✝⁸ : Algebra S R
inst✝⁷ : IsScalarTower S R M
inst✝⁶ : AddCommGroup M₂
inst✝⁵ : Module R M₂
inst✝⁴ : Module S M₂
inst✝³ : SMulCommClass R S M₂
inst✝² : IsScalarTower S R M₂
inst✝¹ : Algebra.IsCentral S R
inst✝ : IsTorsionFree S M₂
f g : M ≃ₗ[R] M₂
h : conjAlgEquiv S f = conjAlgEquiv S g
h✝ : Subsingleton M
x✝ : M
⊢ f x✝ = (1 • g) x✝

Tactic:
;

Goal state after:
R : Type u_1
M : Type u_3
inst✝¹⁶ : Semiring R
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : Module R M
inst✝¹³ : Free R M
S : Type u_4
M₂ : Type u_5
inst✝¹² : CommRing S
inst✝¹¹ : IsCancelMulZero S
inst✝¹⁰ : Module S M
inst✝⁹ : SMulCommClass R S M
inst✝⁸ : Algebra S R
inst✝⁷ : IsScalarTower S R M
inst✝⁶ : AddCommGroup M₂
inst✝⁵ : Module R M₂
inst✝⁴ : Module S M₂
inst✝³ : SMulCommClass R S M₂
inst✝² : IsScalarTower S R M₂
inst✝¹ : Algebra.IsCentral S R
inst✝ : IsTorsionFree S M₂
f g : M ≃ₗ[R] M₂
h : conjAlgEquiv S f = conjAlgEquiv S g
h✝ : Subsingleton M
x✝ : M
⊢ f x✝ = (1 • g) x✝
Module: Mathlib.RingTheory.Algebraic.Basic

Goal state before:
pp: R : Type u
S : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
halg : IsAlgebraic (↥⊥) x
r : R
⊢ ∃ a, (algebraMap R ↥⊥) a = ⟨(Algebra.ofId R S).toRingHom r, ⋯⟩

Tactic:
;

Goal state after:
R : Type u
S : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
halg : IsAlgebraic (↥⊥) x
r : R
⊢ ∃ a, (algebraMap R ↥⊥) a = ⟨(Algebra.ofId R S).toRingHom r, ⋯⟩