mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Data.Nat.Multiplicity

Goal state before:
pp: p : ℕ
hp : Prime p
n b : ℕ
hb : log p (n + 1) < b
⊢ emultiplicity p (n + 1) + emultiplicity p n ! = emultiplicity p n ! + emultiplicity p (n + 1)

Tactic:
add_comm

Goal state after:
p : ℕ
hp : Prime p
n b : ℕ
hb : log p (n + 1) < b
⊢ emultiplicity p n ! + emultiplicity p (n + 1) = emultiplicity p n ! + emultiplicity p (n + 1)
Module: Mathlib.GroupTheory.OrderOfElement

Goal state before:
pp: G : Type u_1
inst✝ : RightCancelMonoid G
x : G
h : ¬IsOfFinOrder x
⊢ {y | ¬IsOfFinOrder y}.Infinite

Tactic:
by
  let s := { n | 0 < n }.image fun n : ℕ => x ^ n
  have hs : s ⊆ { y : G | ¬IsOfFinOrder y } := by
    rintro - ⟨n, hn : 0 < n, rfl⟩ (contra : IsOfFinOrder (x ^ n))
    apply h
    rw [isOfFinOrder_iff_pow_eq_one] at contra ⊢
    obtain ⟨m, hm, hm'⟩ := contra
    exact ⟨n * m, mul_pos hn hm, by rwa [pow_mul]⟩
  suffices s.Infinite by exact this.mono hs
  contrapose! h
  have : ¬Function.Injective fun n : ℕ => x ^ n := by
    have := Set.not_injOn_infinite_finite_image (Set.Ioi_infinite 0) h
    contrapose this
    exact Set.injOn_of_injective this
  rwa [injective_pow_iff_not_isOfFinOrder, Classical.not_not] at this

Goal state after:
Module: Mathlib.GroupTheory.OrderOfElement

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
x y : G
α : Type u_6
inst✝ : MulAction G α
hx : x ∈ zpowers y
a : α
hs : y • a = a
⊢ x • a = a

Tactic:
obtain ⟨k, rfl⟩ := Subgroup.mem_zpowers_iff.mp hx

Goal state after:
G : Type u_1
inst✝¹ : Group G
y : G
α : Type u_6
inst✝ : MulAction G α
a : α
hs : y • a = a
k : ℤ
hx : y ^ k ∈ zpowers y
⊢ y ^ k • a = a
Module: Mathlib.GroupTheory.OrderOfElement

Goal state before:
pp: case refine_1
G : Type u_1
inst✝ : Group G
x : G
⊢ (Injective fun n ↦ x ^ n) → ¬∃ n, 0 < n ∧ x ^ n = 1

Tactic:
rintro h ⟨n, hn, hx⟩

Goal state after:
case refine_1
G : Type u_1
inst✝ : Group G
x : G
h : Injective fun n ↦ x ^ n
n : ℕ
hn : 0 < n
hx : x ^ n = 1
⊢ False
Module: Mathlib.Data.Nat.Multiplicity

Goal state before:
pp: case neg.false
h2 : _root_.Prime 2
n : ℕ
ih : n ≠ 0 → emultiplicity 2 n ! < ↑n
hn : ¬n = 0
this : emultiplicity 2 (2 * n)! < ↑(2 * n)
h : bit false n ≠ 0
⊢ emultiplicity 2 (bit false n)! < ↑(bit false n)

Tactic:
simpa

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

Goal state before:
pp: R : Type u
n : ℕ
inst✝ : Semiring R
p : R[X]
⊢ p.divX.coeff n = p.coeff (n + 1)

Tactic:
add_comm

Goal state after:
R : Type u
n : ℕ
inst✝ : Semiring R
p : R[X]
⊢ p.divX.coeff n = p.coeff (1 + n)
Module: Mathlib.Algebra.Polynomial.Inductions

Goal state before:
pp: case φ_mon
R : Type u
inst✝ : Semiring R
p : R[X]
⊢ ∀ (n : ℕ) (c : R), c ≠ 0 → (divX_hom ((monomial n) c)).natDegree = n - 1

Tactic:
· intro n c c0
    rw [← C_mul_X_pow_eq_monomial, divX_hom_toFun, divX_C_mul, divX_X_pow]
    split_ifs with n0
    · simp [n0]
    · exact natDegree_C_mul_X_pow (n - 1) c c0

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

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p : R[X]
hp0 : p ≠ 0
this : Nontrivial R
h : ¬p.degree ≤ 0
hXp0 : p.divX ≠ 0
⊢ p.divX.leadingCoeff * X.leadingCoeff ≠ 0

Tactic:
simpa

Goal state after:
Module: Mathlib.GroupTheory.OrderOfElement

Goal state before:
pp: G✝ : Type u_1
H : Type u_2
A : Type u_3
α : Type u_4
β : Type u_5
G : Type u_6
inst✝¹ : Group G
inst✝ : Fintype G
S : Set G
a : G
ha : a ∈ S
⊢ 1 ∈ S ^ card G

Tactic:
← pow_card_eq_one

Goal state after:
G✝ : Type u_1
H : Type u_2
A : Type u_3
α : Type u_4
β : Type u_5
G : Type u_6
inst✝¹ : Group G
inst✝ : Fintype G
S : Set G
a : G
ha : a ∈ S
⊢ ?m.40 ^ card G ∈ S ^ card G

G✝ : Type u_1
H : Type u_2
A : Type u_3
α : Type u_4
β : Type u_5
G : Type u_6
inst✝¹ : Group G
inst✝ : Fintype G
S : Set G
a : G
ha : a ∈ S
⊢ G
Module: Mathlib.Algebra.Polynomial.Inductions

Goal state before:
pp: case refine_2.inr.inl
R : Type u
inst✝ : Semiring R
M : R[X] → Prop
f : R[X]
f0 : f.natDegree ≠ 0
h_C_add : ∀ {a : R} {p : R[X]}, M p → M (C a + p)
h_add : ∀ {p q : R[X]}, M p → M q → M (p + q)
h_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M ((monomial n) a)
p q : R[X]
hp : M p
hq : q.natDegree = 0
⊢ M (p + C (q.coeff 0))

Tactic:
add_comm

Goal state after:
case refine_2.inr.inl
R : Type u
inst✝ : Semiring R
M : R[X] → Prop
f : R[X]
f0 : f.natDegree ≠ 0
h_C_add : ∀ {a : R} {p : R[X]}, M p → M (C a + p)
h_add : ∀ {p q : R[X]}, M p → M q → M (p + q)
h_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M ((monomial n) a)
p q : R[X]
hp : M p
hq : q.natDegree = 0
⊢ M (C (q.coeff 0) + p)
Module: Mathlib.Algebra.Polynomial.Inductions

Goal state before:
pp: case refine_3
R : Type u
inst✝ : Semiring R
M : R[X] → Prop
f : R[X]
f0 : f.natDegree ≠ 0
h_C_add : ∀ {a : R} {p : R[X]}, M p → M (C a + p)
h_add : ∀ {p q : R[X]}, M p → M q → M (p + q)
h_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M ((monomial n) a)
n : ℕ
a : R
a✝ : (C a * X ^ n).natDegree = 0 ∨ M (C a * X ^ n)
⊢ (C a * X ^ (n + 1)).natDegree = 0 ∨ M (C a * X ^ (n + 1))

Tactic:
by_cases a0 : a = 0

Goal state after:
case pos
R : Type u
inst✝ : Semiring R
M : R[X] → Prop
f : R[X]
f0 : f.natDegree ≠ 0
h_C_add : ∀ {a : R} {p : R[X]}, M p → M (C a + p)
h_add : ∀ {p q : R[X]}, M p → M q → M (p + q)
h_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M ((monomial n) a)
n : ℕ
a : R
a✝ : (C a * X ^ n).natDegree = 0 ∨ M (C a * X ^ n)
a0 : a = 0
⊢ (C a * X ^ (n + 1)).natDegree = 0 ∨ M (C a * X ^ (n + 1))

case neg
R : Type u
inst✝ : Semiring R
M : R[X] → Prop
f : R[X]
f0 : f.natDegree ≠ 0
h_C_add : ∀ {a : R} {p : R[X]}, M p → M (C a + p)
h_add : ∀ {p q : R[X]}, M p → M q → M (p + q)
h_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M ((monomial n) a)
n : ℕ
a : R
a✝ : (C a * X ^ n).natDegree = 0 ∨ M (C a * X ^ n)
a0 : ¬a = 0
⊢ (C a * X ^ (n + 1)).natDegree = 0 ∨ M (C a * X ^ (n + 1))
Module: Mathlib.Tactic.ComputeDegree

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
d df dg : ℕ
a b : R
f g : R[X]
h_mul_left✝ : f.natDegree ≤ df
h_mul_right✝ : g.natDegree ≤ dg
h_mul_left : f.coeff df = a
h_mul_right : g.coeff dg = b
ddf : df + dg ≤ d
h : ¬d = df + dg
⊢ df + dg ≠ d

Tactic:
· exact ne_comm.mp h

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

Goal state before:
pp: case neg
R : Type u
inst✝ : CommSemiring R
p : ℕ
hp : 0 < p
f : R[X]
n : ℕ
h : ¬p ∣ n
⊢ (∑ x ∈ f.support, if p * x = n then f.coeff x else 0) = 0

Tactic:
rw [Finset.sum_eq_zero]

Goal state after:
case neg
R : Type u
inst✝ : CommSemiring R
p : ℕ
hp : 0 < p
f : R[X]
n : ℕ
h : ¬p ∣ n
⊢ ∀ x ∈ f.support, (if p * x = n then f.coeff x else 0) = 0
Module: Mathlib.Algebra.Polynomial.RingDivision

Goal state before:
pp: case neg
R : Type u
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hp : ¬p = 0
hq : ¬q = 0
⊢ (p * q).trailingDegree = p.trailingDegree + q.trailingDegree

Tactic:
rw [trailingDegree_eq_natTrailingDegree hp, trailingDegree_eq_natTrailingDegree hq,
    trailingDegree_eq_natTrailingDegree (mul_ne_zero hp hq), natTrailingDegree_mul hp hq]
    apply WithTop.coe_add

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

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
p : ℕ
f : R[X]
hp : 0 < p
⊢ ((expand R p) f).leadingCoeff = f.leadingCoeff

Tactic:
simp_rw [leadingCoeff, natDegree_expand, coeff_expand_mul hp]

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

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
p : R[X]
hp : p ≠ 0
this : Decidable (∃ x, p.IsRoot x)
h : ∃ x, p.IsRoot x
x : R
hx : p.IsRoot x
hpd : 0 < p.degree
hd0 : p /ₘ (X - C x) ≠ 0
wf : (p /ₘ (X - C x)).degree < p.degree
t : Multiset R
htd : ↑t.card ≤ (p /ₘ (X - C x)).degree
htr : ∀ (a : R), count a t = rootMultiplicity a (p /ₘ (X - C x))
hdeg : (X - C x).degree ≤ p.degree
hdiv0 : p /ₘ (X - C x) ≠ 0
⊢ ↑t.card + 1 ≤ 1 + (p /ₘ (X - C x)).degree

Tactic:
add_comm

Goal state after:
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
p : R[X]
hp : p ≠ 0
this : Decidable (∃ x, p.IsRoot x)
h : ∃ x, p.IsRoot x
x : R
hx : p.IsRoot x
hpd : 0 < p.degree
hd0 : p /ₘ (X - C x) ≠ 0
wf : (p /ₘ (X - C x)).degree < p.degree
t : Multiset R
htd : ↑t.card ≤ (p /ₘ (X - C x)).degree
htr : ∀ (a : R), count a t = rootMultiplicity a (p /ₘ (X - C x))
hdeg : (X - C x).degree ≤ p.degree
hdiv0 : p /ₘ (X - C x) ≠ 0
⊢ 1 + ↑t.card ≤ 1 + (p /ₘ (X - C x)).degree
Module: Mathlib.Algebra.Polynomial.RingDivision

Goal state before:
pp: case pos
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
p : R[X]
hp : p ≠ 0
this : Decidable (∃ x, p.IsRoot x)
h : ∃ x, p.IsRoot x
x : R
hx : p.IsRoot x
hpd : 0 < p.degree
hd0 : p /ₘ (X - C x) ≠ 0
wf : (p /ₘ (X - C x)).degree < p.degree
t : Multiset R
htd : ↑t.card ≤ (p /ₘ (X - C x)).degree
htr : ∀ (a : R), count a t = rootMultiplicity a (p /ₘ (X - C x))
hdeg : (X - C x).degree ≤ p.degree
hdiv0 : p /ₘ (X - C x) ≠ 0
a : R
ha : a = x
⊢ count x t + 1 = 1 + count x t

Tactic:
add_comm

Goal state after:
case pos
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
p : R[X]
hp : p ≠ 0
this : Decidable (∃ x, p.IsRoot x)
h : ∃ x, p.IsRoot x
x : R
hx : p.IsRoot x
hpd : 0 < p.degree
hd0 : p /ₘ (X - C x) ≠ 0
wf : (p /ₘ (X - C x)).degree < p.degree
t : Multiset R
htd : ↑t.card ≤ (p /ₘ (X - C x)).degree
htr : ∀ (a : R), count a t = rootMultiplicity a (p /ₘ (X - C x))
hdeg : (X - C x).degree ≤ p.degree
hdiv0 : p /ₘ (X - C x) ≠ 0
a : R
ha : a = x
⊢ 1 + count x t = 1 + count x t
Module: Mathlib.Algebra.Polynomial.Derivative

Goal state before:
pp: R : Type u
inst✝ : Semiring R
n k : ℕ
f : R[X]
⊢ (⇑derivative)^[k] (↑n * f) = ↑n * (⇑derivative)^[k] f

Tactic:
by
  induction k generalizing f <;> simp [*]

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

Goal state before:

Tactic:
(derivative^[k + 1] p).coeff m
        _ = Nat.descFactorial (Nat.succ (m + k)) k • p.coeff (m + k.succ) * (m + 1) := by
          rw [Function.iterate_succ_apply', coeff_derivative, ih m.succ, Nat.succ_add, Nat.add_succ]
        _ = ((m + 1) * Nat.descFactorial (Nat.succ (m + k)) k) • p.coeff (m + k.succ) := by
          rw [← Nat.cast_add_one, ← nsmul_eq_mul', smul_smul]
        _ = Nat.descFactorial (m.succ + k) k.succ • p.coeff (m + k.succ) := by
          rw [← Nat.succ_add, Nat.descFactorial_succ, add_tsub_cancel_right]
        _ = Nat.descFactorial (m + k.succ) k.succ • p.coeff (m + k.succ) := by
          rw [Nat.succ_add_eq_add_succ]

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

Goal state before:
pp: case inr.refine_1.e_a.refine_2
R : Type u
inst✝ : Ring R
p : R[X]
a : R
n✝ : ℕ
this :
  ∀ (n : ℕ), p.natDegree ≤ n → (p /ₘ (X - C a)).coeff n = ∑ i ∈ Icc (n + 1) p.natDegree, a ^ (i - (n + 1)) * p.coeff i
h : ¬p.natDegree ≤ n✝
n : ℕ
hn : n < p.natDegree
x✝ : n✝ ≤ n
ih : (p /ₘ (X - C a)).coeff (n + 1) = ∑ i ∈ Icc (n + 1 + 1) p.natDegree, a ^ (i - (n + 1 + 1)) * p.coeff i
i : ℕ
hi : i ∈ Icc (n + 1 + 1) p.natDegree
⊢ 1 + (n + 1) ≤ i

Tactic:
add_comm

Goal state after:
case inr.refine_1.e_a.refine_2
R : Type u
inst✝ : Ring R
p : R[X]
a : R
n✝ : ℕ
this :
  ∀ (n : ℕ), p.natDegree ≤ n → (p /ₘ (X - C a)).coeff n = ∑ i ∈ Icc (n + 1) p.natDegree, a ^ (i - (n + 1)) * p.coeff i
h : ¬p.natDegree ≤ n✝
n : ℕ
hn : n < p.natDegree
x✝ : n✝ ≤ n
ih : (p /ₘ (X - C a)).coeff (n + 1) = ∑ i ∈ Icc (n + 1 + 1) p.natDegree, a ^ (i - (n + 1 + 1)) * p.coeff i
i : ℕ
hi : i ∈ Icc (n + 1 + 1) p.natDegree
⊢ n + 1 + 1 ≤ i
Module: Mathlib.Order.Filter.Finite

Goal state before:
pp: case mp.superset
α : Type u
s : Set (Set α)
U s✝ t✝ : Set α
a✝ : GenerateSets s s✝
hVW : s✝ ⊆ t✝
hV : ∃ t ⊆ s, t.Finite ∧ ⋂₀ t ⊆ s✝
⊢ ∃ t ⊆ s, t.Finite ∧ ⋂₀ t ⊆ t✝

Tactic:
rcases hV with ⟨t, hts, ht, htV⟩

Goal state after:
case mp.superset
α : Type u
s : Set (Set α)
U s✝ t✝ : Set α
a✝ : GenerateSets s s✝
hVW : s✝ ⊆ t✝
t : Set (Set α)
hts : t ⊆ s
ht : t.Finite
htV : ⋂₀ t ⊆ s✝
⊢ ∃ t ⊆ s, t.Finite ∧ ⋂₀ t ⊆ t✝
Module: Mathlib.Order.Filter.CountablyGenerated

Goal state before:
pp: α : Type u_1
ι' : Sort u_5
p : ι' → Prop
s✝ : ι' → Set α
s : Set (Set α)
hsc : s.Countable
hs : (generate s).HasBasis p s✝
⊢ ∃ x, ⨅ s_1 ∈ s, 𝓟 s_1 = ⨅ i, 𝓟 (x i)

Tactic:
exact countable_biInf_principal_eq_seq_iInf hsc

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

Goal state before:
pp: R : Type u
inst✝ : CommRing R
p₁ p₂ q : R[X]
hq : q.Monic
a✝ : Nontrivial R
f : R[X]
sub_eq : p₁ - p₂ = q * f
⊢ p₂ + q * f = q * f + p₂

Tactic:
add_comm

Goal state after:
R : Type u
inst✝ : CommRing R
p₁ p₂ q : R[X]
hq : q.Monic
a✝ : Nontrivial R
f : R[X]
sub_eq : p₁ - p₂ = q * f
⊢ q * f + p₂ = q * f + p₂
Module: Mathlib.Order.Filter.Ker

Goal state before:
pp: ι : Sort u_1
α : Type u_2
f : ι → Filter α
x : α
s : ι → Set α
hsf : ∀ (i : ι), s i ∈ f i
hxs : ∀ (i : ι), x ∉ s i
⊢ x ∉ ⋃ i, s i

Tactic:
simpa

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

Goal state before:
pp: case zero
R : Type u
inst✝ : CommSemiring R
q : R[X]
n : ℕ
p : R[X]
dvd : q ^ n ∣ p
⊢ q ^ (n - 0) ∣ (⇑derivative)^[0] p

Tactic:
simpa

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

Goal state before:
pp: R : Type u
inst✝ : CommRing R
p q : R[X]
hmonic : q.Monic
hdegree : q.degree ≤ p.degree
a✝ : Nontrivial R
h : q.leadingCoeff * (p /ₘ q).leadingCoeff ≠ 0
⊢ (p /ₘ q).leadingCoeff = (p %ₘ q + q * (p /ₘ q)).leadingCoeff

Tactic:
rw [leadingCoeff_add_of_degree_lt, leadingCoeff_monic_mul hmonic]

Goal state after:
R : Type u
inst✝ : CommRing R
p q : R[X]
hmonic : q.Monic
hdegree : q.degree ≤ p.degree
a✝ : Nontrivial R
h : q.leadingCoeff * (p /ₘ q).leadingCoeff ≠ 0
⊢ (p %ₘ q).degree < (q * (p /ₘ q)).degree
Module: Mathlib.Algebra.Polynomial.Derivative

Goal state before:
pp: case e_a
R : Type u
ι : Type y
inst✝¹ : CommSemiring R
inst✝ : DecidableEq ι
s✝ : Multiset ι
f : ι → R[X]
i : ι
s : Multiset ι
h :
  derivative (Multiset.map f s).prod =
    (Multiset.map (fun i ↦ (Multiset.map f (s.erase i)).prod * derivative (f i)) s).sum
⊢ f i * (Multiset.map (fun i ↦ (Multiset.map f (s.erase i)).prod * derivative (f i)) s).sum =
    (Multiset.map (fun i_1 ↦ (Multiset.map f ((i ::ₘ s).erase i_1)).prod * derivative (f i_1)) s).sum

Tactic:
← AddMonoidHom.coe_mulLeft,

Goal state after:
case e_a
R : Type u
ι : Type y
inst✝¹ : CommSemiring R
inst✝ : DecidableEq ι
s✝ : Multiset ι
f : ι → R[X]
i : ι
s : Multiset ι
h :
  derivative (Multiset.map f s).prod =
    (Multiset.map (fun i ↦ (Multiset.map f (s.erase i)).prod * derivative (f i)) s).sum
⊢ (AddMonoidHom.mulLeft (f i)) (Multiset.map (fun i ↦ (Multiset.map f (s.erase i)).prod * derivative (f i)) s).sum =
    (Multiset.map (fun i_1 ↦ (Multiset.map f ((i ::ₘ s).erase i_1)).prod * derivative (f i_1)) s).sum
Module: Mathlib.Algebra.Polynomial.Derivative

Goal state before:
pp: R : Type u
inst✝ : Ring R
n : ℤ
k : ℕ
f : R[X]
⊢ (⇑derivative)^[k] (↑n * f) = ↑n * (⇑derivative)^[k] f

Tactic:
by
  induction k generalizing f <;> simp [*]

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

Goal state before:
pp: R : Type u
inst✝ : CommRing R
S : Finset R
k : ℕ
ind : (⇑derivative)^[k] (∏ a ∈ S, (X - C a)) = ↑k ! * ∑ T ∈ powersetCard (#S - k) S, ∏ a ∈ T, (X - C a)
hk : k + 1 ≤ #S
⊢ (⇑derivative)^[k + 1] (∏ a ∈ S, (X - C a)) = ↑(k + 1)! * ∑ T ∈ powersetCard (#S - (k + 1)) S, ∏ a ∈ T, (X - C a)

Tactic:
add_comm

Goal state after:
R : Type u
inst✝ : CommRing R
S : Finset R
k : ℕ
ind : (⇑derivative)^[k] (∏ a ∈ S, (X - C a)) = ↑k ! * ∑ T ∈ powersetCard (#S - k) S, ∏ a ∈ T, (X - C a)
hk : k + 1 ≤ #S
⊢ (⇑derivative)^[1 + k] (∏ a ∈ S, (X - C a)) = ↑(k + 1)! * ∑ T ∈ powersetCard (#S - (k + 1)) S, ∏ a ∈ T, (X - C a)
Module: Mathlib.Order.Filter.TendstoCofinite

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
f : α → β
inst✝ : TendstoCofinite f
x : β →₀ ℕ
s : Finset α := x.support.sup fun t ↦ ⋯.toFinset
e : ↥s ↪ α := Function.Embedding.subtype fun u ↦ u ∈ s
y : α →₀ ℕ
hy : mapDomain f y = x
⊢ ↑y.support ⊆ range ⇑e

Tactic:
suffices y.support ⊆ s by simpa [e]
    simpa [← hy, mapDomain, sum, Finset.subset_iff, single_apply, s] using
      fun i hi ↦ ⟨i, by simp [hi]⟩

Goal state after:
Module: Mathlib.Algebra.BigOperators.Associated

Goal state before:
pp: M : Type u_2
inst✝ : CommMonoid M
p q : Multiset (Associates M)
h : p ≤ q
this✝ : DecidableEq (Associates M)
this : p.prod * 1 ≤ p.prod * (q - p).prod
⊢ p.prod ≤ (p + (q - p)).prod

Tactic:
simpa

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

Goal state before:
pp: R : Type
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hp : p.natDegree ≠ 0
x : R
hx : ((fun x ↦ eval x p) ⁻¹' {x}).Infinite
⊢ {x_1 | eval x_1 p = eval x_1 (C x)}.Infinite

Tactic:
simpa

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

Goal state before:
pp: R : Type u
a : R
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
ha : a ≠ 0
⊢ (a • p).roots = p.roots

Tactic:
smul_eq_C_mul,

Goal state after:
R : Type u
a : R
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
ha : a ≠ 0
⊢ (C a * p).roots = p.roots
Module: Mathlib.Data.Finsupp.Weight

Goal state before:
pp: σ : Type u_5
s t : Set ℕ
f : σ →₀ ℕ
m : ℕ
hm : m ∈ s
n : ℕ
hn : n ∈ t
e : m + n = degree f
⊢ f ∈ ⇑degree ⁻¹' s + ⇑degree ⁻¹' t

Tactic:
obtain ⟨g, hgf, rfl⟩ := exists_le_degree_eq f m (by grind)

Goal state after:
σ : Type u_5
s t : Set ℕ
f : σ →₀ ℕ
n : ℕ
hn : n ∈ t
g : σ →₀ ℕ
hgf : g ≤ f
hm : degree g ∈ s
e : degree g + n = degree f
⊢ f ∈ ⇑degree ⁻¹' s + ⇑degree ⁻¹' t
Module: Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors

Goal state before:
pp: case inr.cons
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
h✝ : Nontrivial α
a✝ : α
s✝ : Multiset α
IH : 0 ∉ s✝ → normalizedFactors s✝.prod = (Multiset.map normalizedFactors s✝).sum
hs : 0 ∉ a✝ ::ₘ s✝
⊢ 0 ∉ s✝

Tactic:
exact fun h ↦ hs (Multiset.mem_cons_of_mem h)

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

Goal state before:
pp: case inr.cons.hy
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
h✝ : Nontrivial α
a✝ : α
s✝ : Multiset α
IH : 0 ∉ s✝ → normalizedFactors s✝.prod = (Multiset.map normalizedFactors s✝).sum
hs : 0 ∉ a✝ ::ₘ s✝
⊢ 0 ∉ s✝

Tactic:
exact fun h ↦ hs (Multiset.mem_cons_of_mem h)

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

Goal state before:
pp: S : Type v
T : Type w
inst✝⁵ : CommRing T
inst✝⁴ : IsDomain T
inst✝³ : CommRing S
inst✝² : IsDomain S
inst✝¹ : Algebra T S
inst✝ : Module.IsTorsionFree T S
a : T
p : T[X]
ha : a ≠ 0
⊢ (a • p).aroots S = p.aroots S

Tactic:
smul_eq_C_mul,

Goal state after:
S : Type v
T : Type w
inst✝⁵ : CommRing T
inst✝⁴ : IsDomain T
inst✝³ : CommRing S
inst✝² : IsDomain S
inst✝¹ : Algebra T S
inst✝ : Module.IsTorsionFree T S
a : T
p : T[X]
ha : a ≠ 0
⊢ (C a * p).aroots S = p.aroots S
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: S : Type v
T : Type w
inst✝⁵ : CommRing T
inst✝⁴ : IsDomain T
inst✝³ : CommRing S
inst✝² : IsDomain S
inst✝¹ : Algebra T S
inst✝ : Module.IsTorsionFree T S
a : T
ha : a ≠ 0
n : ℕ
⊢ (X ^ n).aroots S = n • {0}

Tactic:
aroots_X_pow

Goal state after:
S : Type v
T : Type w
inst✝⁵ : CommRing T
inst✝⁴ : IsDomain T
inst✝³ : CommRing S
inst✝² : IsDomain S
inst✝¹ : Algebra T S
inst✝ : Module.IsTorsionFree T S
a : T
ha : a ≠ 0
n : ℕ
⊢ n • {0} = n • {0}
Module: Mathlib.RingTheory.Polynomial.Content

Goal state before:
pp: case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
p : R[X]
h : ¬p.coeff p.natDegree = 0
⊢ p.content = gcd p.leadingCoeff p.eraseLead.content

Tactic:
← Ne,

Goal state after:
case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
p : R[X]
h : p.coeff p.natDegree ≠ 0
⊢ p.content = gcd p.leadingCoeff p.eraseLead.content
Module: Mathlib.RingTheory.Polynomial.Content

Goal state before:
pp: case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
p : R[X]
h : p.coeff p.natDegree ≠ 0
⊢ p.content = gcd p.leadingCoeff p.eraseLead.content

Tactic:
← mem_support_iff

Goal state after:
case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
p : R[X]
h : p.natDegree ∈ p.support
⊢ p.content = gcd p.leadingCoeff p.eraseLead.content
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:
Subtype.coe_mk

Goal state after:
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'
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:
Subtype.coe_mk

Goal state after:
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'
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:
grw [mul_assoc, associated_content_C_mul, associated_content_C_mul, mul_comm p.primPart,
        mul_assoc, associated_content_C_mul, associated_content_C_mul, h, mul_one,
        content_primPart, content_primPart, mul_one, mul_one]

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

Goal state before:
pp: case pos
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 : IsUnit (r.coeff 0)
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.coeff 0 ∣ s.content
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : r.natDegree ≤ s.natDegree
sC : s.natDegree ≤ 0
⊢ False

Tactic:
apply rs rprim.dvd

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

Goal state before:
pp: R : Type u
inst✝¹ : Field R
p : R[X]
inst✝ : DecidableEq R
hp0 : ¬p.leadingCoeff = 0
⊢ (normalize p).Monic

Tactic:
← Ne,

Goal state after:
R : Type u
inst✝¹ : Field R
p : R[X]
inst✝ : DecidableEq R
hp0 : p.leadingCoeff ≠ 0
⊢ (normalize p).Monic
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝¹ : Field R
p : R[X]
inst✝ : DecidableEq R
hp0 : p ≠ 0
⊢ (normalize p).Monic

Tactic:
by
  rw [Ne, ← leadingCoeff_eq_zero, ← Ne, ← isUnit_iff_ne_zero] at hp0
  rw [Monic, leadingCoeff_normalize, normalize_eq_one]
  apply hp0

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
k : ℕ
⊢ p ^ k = ↑⟨p, hp⟩ ^ k

Tactic:
Subtype.coe_mk

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 = p ^ k