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

Goal state before:
pp: R : Type u
inst✝¹ : CommSemiring R
S : Type v
inst✝ : CommSemiring S
p : ℕ
hp : p ≠ 0
f : R →+* S
q : R[X]
n : ℕ
⊢ (map f (contract p q)).coeff n = (contract p (map f q)).coeff n

Tactic:
by
  simp only [coeff_map, coeff_contract hp]

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

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p : R[X]
n : ℕ
h : n ∈ (derivative p).support
h1 : p.coeff (n + 1) = 0
⊢ 0 * (↑n + 1) = 0

Tactic:
zero_mul

Goal state after:
R : Type u
inst✝ : Semiring R
p : R[X]
n : ℕ
h : n ∈ (derivative p).support
h1 : p.coeff (n + 1) = 0
⊢ 0 = 0
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝ : Ring R
p q : R[X]
this✝ : DecidableEq R := Classical.decEq R
hp0 : ¬p = 0
hq : q.Monic
h : q.degree ≤ p.degree
this : Nontrivial R
⊢ (p /ₘ q).degree ≤ ↑q.natDegree + (p /ₘ q).degree

Tactic:
degree_eq_natDegree (mt (divByMonic_eq_zero_iff hq).1 (not_lt.2 h))

Goal state after:
R : Type u
inst✝ : Ring R
p q : R[X]
this✝ : DecidableEq R := Classical.decEq R
hp0 : ¬p = 0
hq : q.Monic
h : q.degree ≤ p.degree
this : Nontrivial R
⊢ ↑(p /ₘ q).natDegree ≤ ↑q.natDegree + ↑(p /ₘ q).natDegree
Module: Mathlib.Algebra.Polynomial.Expand

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
p n : ℕ
inst✝ : ExpChar R p
f : R[X]
r : R
h0 : f ≠ 0
g : R[X]
hg : f = (X - C (r ^ p ^ n)) ^ rootMultiplicity (r ^ p ^ n) f * g
ndvd : ¬((expand R (p ^ n)) g).IsRoot r
| rootMultiplicity r ((X ^ p ^ n - C r ^ p ^ n) ^ rootMultiplicity (r ^ p ^ n) f * (expand R (p ^ n)) g)

Tactic:
← sub_pow_expChar_pow,

Goal state after:
R : Type u
inst✝¹ : CommRing R
p n : ℕ
inst✝ : ExpChar R p
f : R[X]
r : R
h0 : f ≠ 0
g : R[X]
hg : f = (X - C (r ^ p ^ n)) ^ rootMultiplicity (r ^ p ^ n) f * g
ndvd : ¬((expand R (p ^ n)) g).IsRoot r
| rootMultiplicity r (((X - C r) ^ p ^ n) ^ rootMultiplicity (r ^ p ^ n) f * (expand R (p ^ n)) g)
Module: Mathlib.Algebra.Polynomial.Derivative

Goal state before:
pp: R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
p : R[X]
f : R →+* S
n✝ : ℕ := max p.natDegree (map f p).natDegree
n : ℕ
⊢ 0 * X ^ (n - 1) = 0

Tactic:
zero_mul

Goal state after:
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
p : R[X]
f : R →+* S
n✝ : ℕ := max p.natDegree (map f p).natDegree
n : ℕ
⊢ 0 = 0
Module: Mathlib.RingTheory.Polynomial.Tower

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁹ : CommSemiring R
inst✝⁸ : CommSemiring A
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra A B
inst✝⁴ : Algebra R B
inst✝³ : IsScalarTower R A B
inst✝² : IsDomain A
inst✝¹ : IsTorsionFree A B
inst✝ : Nontrivial B
x : A
p : R[X]
⊢ (aeval ((algebraMap A B) x)) p = 0 ↔ (aeval x) p = 0

Tactic:
aeval_algebraMap_apply,

Goal state after:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁹ : CommSemiring R
inst✝⁸ : CommSemiring A
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra A B
inst✝⁴ : Algebra R B
inst✝³ : IsScalarTower R A B
inst✝² : IsDomain A
inst✝¹ : IsTorsionFree A B
inst✝ : Nontrivial B
x : A
p : R[X]
⊢ (algebraMap A B) ((aeval x) p) = 0 ↔ (aeval x) p = 0
Module: Mathlib.RingTheory.Polynomial.Tower

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Semiring B
inst✝³ : Algebra R A
inst✝² : Algebra A B
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R A B
x : A
p : R[X]
h : Function.Injective ⇑(algebraMap A B)
⊢ (aeval ((algebraMap A B) x)) p = 0 ↔ (aeval x) p = 0

Tactic:
aeval_algebraMap_apply,

Goal state after:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Semiring B
inst✝³ : Algebra R A
inst✝² : Algebra A B
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R A B
x : A
p : R[X]
h : Function.Injective ⇑(algebraMap A B)
⊢ (algebraMap A B) ((aeval x) p) = 0 ↔ (aeval x) p = 0
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝ : Ring R
f g q r : R[X]
hg : g.Monic
h : r + g * q = f ∧ r.degree < g.degree
a✝ : Nontrivial R
h₁ : r - f %ₘ g = -g * (q - f /ₘ g)
h₂ : (r - f %ₘ g).degree = (g * (q - f /ₘ g)).degree
h₄ : (r - f %ₘ g).degree < g.degree
h₅ : q - f /ₘ g = 0
⊢ r - f %ₘ g = 0

Tactic:
by simpa [h₅] using h₁

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

Goal state before:
pp: R : Type u
inst✝ : Ring R
p q : R[X]
hq : q.Monic
h✝ : q ∣ p
a✝ : Nontrivial R
r : R[X]
hr : p = q * r
hpq0 : ¬p %ₘ q = 0
hmod : p %ₘ q = q * (r - p /ₘ q)
this : (q * (r - p /ₘ q)).degree < q.degree
h : (r - p /ₘ q).leadingCoeff = 0
⊢ (q * (r - p /ₘ q)).leadingCoeff = 0

Tactic:
leadingCoeff_eq_zero.1 h,

Goal state after:
R : Type u
inst✝ : Ring R
p q : R[X]
hq : q.Monic
h✝ : q ∣ p
a✝ : Nontrivial R
r : R[X]
hr : p = q * r
hpq0 : ¬p %ₘ q = 0
hmod : p %ₘ q = q * (r - p /ₘ q)
this : (q * (r - p /ₘ q)).degree < q.degree
h : (r - p /ₘ q).leadingCoeff = 0
⊢ (q * 0).leadingCoeff = 0
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
a : R
inst✝ : CommRing R
p : R[X]
h : (X - C a) * (p /ₘ (X - C a)) = p
⊢ 0 * eval a (p /ₘ (X - C a)) = 0

Tactic:
zero_mul

Goal state after:
R : Type u
a : R
inst✝ : CommRing R
p : R[X]
h : (X - C a) * (p /ₘ (X - C a)) = p
⊢ 0 = 0
Module: Mathlib.Order.Filter.Pi

Goal state before:
pp: case mpr
ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
s : (i : ι) → Set (α i)
i : ι
hi : (s i)ᶜ ∈ f i
⊢ (univ.pi s)ᶜ ∈ pi f

Tactic:
filter_upwards [mem_pi_of_mem i hi] with x using mt fun h => h i trivial

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

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
p q : R[X]
r : R
⊢ 0 = 0 * derivative q

Tactic:
zero_mul

Goal state after:
R : Type u
inst✝ : CommSemiring R
p q : R[X]
r : R
⊢ 0 = 0
Module: Mathlib.Order.Heyting.Boundary

Goal state before:
pp: α : Type u_1
inst✝ : CoheytingAlgebra α
a : α
⊢ ￢￢a \ ∂ a ⊔ ⊥ = ￢￢a

Tactic:
sup_bot_eq,

Goal state after:
α : Type u_1
inst✝ : CoheytingAlgebra α
a : α
⊢ ￢￢a \ ∂ a = ￢￢a
Module: Mathlib.Order.Filter.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
f : α × β → γ
x : Filter α
y : Filter β
z : Filter γ
⊢ Tendsto f (x ×ˢ y) z ↔ ∀ W ∈ z, ∃ U ∈ x, ∃ V ∈ y, ∀ (x : α) (y : β), x ∈ U → y ∈ V → f (x, y) ∈ W

Tactic:
by
  simp only [tendsto_def, mem_prod_iff, prod_sub_preimage_iff]

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

Goal state before:
pp: case pos
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
j : ι
hj : j ∈ s
hij : i = j
⊢ (Multiset.map f (i ::ₘ s.erase j)).prod * derivative (f j) =
    (Multiset.map f ((i ::ₘ s).erase j)).prod * derivative (f j)

Tactic:
simp [hij, Multiset.cons_erase hj]

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

Goal state before:
pp: R : Type u
inst✝ : CommRing R
k : ℕ
ind : ∀ {P : R[X]}, P.degree < ↑k → (⇑derivative)^[k] P = 0
P : R[X]
h : P.degree < ↑(k + 1)
hP : ¬P = 0
hP' : ¬derivative P = 0
⊢ P.natDegree ≠ 0

Tactic:
contrapose hP'

Goal state after:
R : Type u
inst✝ : CommRing R
k : ℕ
ind : ∀ {P : R[X]}, P.degree < ↑k → (⇑derivative)^[k] P = 0
P : R[X]
h : P.degree < ↑(k + 1)
hP : ¬P = 0
hP' : P.natDegree = 0
⊢ derivative P = 0
Module: Mathlib.Order.Filter.TendstoCofinite

Goal state before:
pp: α : Type u_1
β : Type u_2
M : Type u_5
f : α → β
inst✝¹ : AddCommMonoid M
inst✝ : TendstoCofinite f
u v : α → M
⊢ mapDomain f (u + v) = mapDomain f u + mapDomain f v

Tactic:
ext; simp [mapDomain, Finset.sum_add_distrib]

Goal state after:
Module: Mathlib.Data.Finsupp.Weight

Goal state before:
pp: σ : Type u_1
M : Type u_2
w : σ → M
inst✝ : AddCommMonoid M
f : σ →₀ ℕ
i : σ
hi : f i ≠ 0
⊢ (weight w) (f - single i 1) + w i = (weight w) f

Tactic:
conv_rhs => rw [← sub_add_single_one_cancel hi, weight_apply]
  rw [sum_add_index', sum_single_index, one_smul, weight_apply]
  exacts [zero_smul .., fun _ ↦ zero_smul .., fun _ _ _ ↦ add_smul ..]

Goal state after:
Module: Mathlib.Data.Finsupp.Weight

Goal state before:
pp: σ : Type u_1
R : Type u_5
inst✝ : Semiring R
⊢ degree = weight fun x ↦ 1

Tactic:
by
  ext d
  simp [weight_apply, smul_eq_mul, mul_one]

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

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a b : α
hc : IsRelPrime a b
x : α
hxa : x ∈ normalizedFactors a
hxb : x ∈ normalizedFactors b
⊢ False

Tactic:
have x_dvd_a := dvd_of_mem_normalizedFactors hxa

Goal state after:
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a b : α
hc : IsRelPrime a b
x : α
hxa : x ∈ normalizedFactors a
hxb : x ∈ normalizedFactors b
x_dvd_a : x ∣ a
⊢ False
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic

Goal state before:
pp: case inr
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : α
h : a ~ᵤ b
ha : ¬a = 0
hb : ¬b = 0
⊢ Rel Associated (factors a) (factors b)

Tactic:
· refine factors_unique irreducible_of_factor irreducible_of_factor ?_
    exact ((factors_prod ha).trans h).trans (factors_prod hb).symm

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

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
hn : 0 < n
a x : R
⊢ x ^ n - a = 0 ↔ x ^ n = a

Tactic:
sub_eq_zero

Goal state after:
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
hn : 0 < n
a x : R
⊢ x ^ n = a ↔ x ^ n = a
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: n : ℕ
R : Type u_1
a : R
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
⊢ nthRootsFinset n a = (nthRoots n a).toFinset

Tactic:
unfold nthRootsFinset

Goal state after:
n : ℕ
R : Type u_1
a : R
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
⊢ (nthRoots n a).toFinset = (nthRoots n a).toFinset
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic

Goal state before:
pp: case refine_2.e_a
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
hc : c = 0
⊢ False

Tactic:
refine not_irreducible_zero (hq _ ?_)

Goal state after:
case refine_2.e_a
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
hc : c = 0
⊢ 0 ∈ q
Module: Mathlib.RingTheory.Polynomial.Content

Goal state before:
pp: case mp
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
p : R[X]
h : ∀ x ∈ p.support, p.coeff x = 0
⊢ p = 0

Tactic:
ext n
    simp_all

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

Goal state before:
pp: case neg.inr
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.natDegree + q.natDegree = n
⊢ Associated (p * q).content (p.content * q.content)

Tactic:
← q.natDegree_primPart,

Goal state after:
case neg.inr
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.natDegree + q.primPart.natDegree = n
⊢ Associated (p * q).content (p.content * q.content)
Module: Mathlib.RingTheory.Polynomial.Content

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
inst✝ : StrongNormalizedGCDMonoid R
p q : R[X]
h0 : ¬C (p * q).content = 0
⊢ C p.content * p.primPart * (C q.content * q.primPart) = C (p * q).content * (p.primPart * q.primPart)

Tactic:
rw [content_mul, map_mul]

Goal state after:
R : Type u_2
inst✝¹ : CommRing R
inst✝ : StrongNormalizedGCDMonoid R
p q : R[X]
h0 : ¬C (p * q).content = 0
⊢ C p.content * p.primPart * (C q.content * q.primPart) = C p.content * C q.content * (p.primPart * q.primPart)
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
hα : eval₂ ϕ α (EuclideanDomain.gcd f g) = 0
p : R[X]
hp : f = EuclideanDomain.gcd f g * p
⊢ 0 * eval₂ ϕ α p = 0

Tactic:
zero_mul

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
hα : eval₂ ϕ α (EuclideanDomain.gcd f g) = 0
p : R[X]
hp : f = EuclideanDomain.gcd f g * p
⊢ 0 = 0
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
hα : eval₂ ϕ α (EuclideanDomain.gcd f g) = 0
p : R[X]
hp : g = EuclideanDomain.gcd f g * p
⊢ 0 * eval₂ ϕ α p = 0

Tactic:
zero_mul

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
hα : eval₂ ϕ α (EuclideanDomain.gcd f g) = 0
p : R[X]
hp : g = EuclideanDomain.gcd f g * p
⊢ 0 = 0
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: case mpr
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : α
ha0 : a ≠ 0
hb0 : b ≠ 0
⊢ (∀ {d : α}, d ∣ a → d ∣ b → ¬Prime d) → Associates.mk a ⊓ Associates.mk b = 1

Tactic:
· contrapose
    intro hg hc
    obtain ⟨p, hp, hpa, hpb⟩ := exists_prime_dvd_of_not_inf_one ha0 hb0 hg
    exact hc hpa hpb hp

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
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
⊢ Associated ((C r.leadingCoeff).primPart * s.primPart) s.primPart

Tactic:
exact associated_unit_mul_left _ _ (isUnit_primPart_C _)

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

Goal state before:
pp: case neg
R : Type u
inst✝ : Field R
x : R
H : ¬x = 0
⊢ ¬Irreducible (C x)

Tactic:
exact fun hx => hx.not_isUnit <| isUnit_C.2 <| isUnit_iff_ne_zero.2 H

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

Goal state before:
pp: R : Type u
inst✝ : Field R
p : R[X]
hp0 : p ≠ 0
hpu : ¬IsUnit p
⊢ Irreducible p ↔ ∀ (q : R[X]), q.degree ≤ ↑(p.natDegree / 2) → q ∣ p → IsUnit q

Tactic:
by
  rw [← irreducible_mul_leadingCoeff_inv,
      (monic_mul_leadingCoeff_inv hp0).irreducible_iff_degree_lt]
  · simp [hp0, natDegree_mul_leadingCoeff_inv]
  · contrapose hpu
    exact .of_mul_eq_one _ hpu

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

Tactic:
rw [or_iff_not_imp_left, ← Ne]

Goal state after:
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 : Associates α
ha : a ≠ 0
b : Associates α
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
p : Associates α
hp : Irreducible p
⊢ p.count a.factors ≠ 0 → p.count b.factors = 0
Module: Mathlib.RingTheory.Algebraic.Basic

Goal state before:
pp: R : Type u
S : Type u_1
A : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Ring A
inst✝³ : Algebra R A
inst✝² : Algebra R S
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
a : S
x✝ : IsAlgebraic R a
f : R[X]
hf₁ : f ≠ 0
hf₂ : (aeval a) f = 0
⊢ (aeval ((algebraMap S A) a)) f = 0

Tactic:
aeval_algebraMap_apply,

Goal state after:
R : Type u
S : Type u_1
A : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Ring A
inst✝³ : Algebra R A
inst✝² : Algebra R S
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
a : S
x✝ : IsAlgebraic R a
f : R[X]
hf₁ : f ≠ 0
hf₂ : (aeval a) f = 0
⊢ (algebraMap S A) ((aeval a) f) = 0
Module: Mathlib.Algebra.Central.End

Goal state before:
pp: R : Type u_1
S : Type u_2
M : Type u_3
inst✝¹⁴ : Semiring R
inst✝¹³ : AddCommMonoid M
inst✝¹² : Module R M
inst✝¹¹ : Free R M
inst✝¹⁰ : CommSemiring S
inst✝⁹ : Module S M
inst✝⁸ : SMulCommClass R S M
inst✝⁷ : Algebra S R
inst✝⁶ : IsScalarTower S R M
M₂ : Type u_4
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
inst✝³ : Module S M₂
inst✝² : SMulCommClass R S M₂
inst✝¹ : IsScalarTower S R M₂
inst✝ : Algebra.IsCentral S R
f g : M ≃ₗ[R] M₂
⊢ (∀ (a : End R M), a ∘ₗ ↑g.symm ∘ₗ ↑f = ↑g.symm ∘ₗ ↑f ∘ₗ a) ↔ ∃ α, ⇑f = α • ⇑g

Tactic:
← comp_assoc _ _ g.symm.toLinearMap,

Goal state after:
R : Type u_1
S : Type u_2
M : Type u_3
inst✝¹⁴ : Semiring R
inst✝¹³ : AddCommMonoid M
inst✝¹² : Module R M
inst✝¹¹ : Free R M
inst✝¹⁰ : CommSemiring S
inst✝⁹ : Module S M
inst✝⁸ : SMulCommClass R S M
inst✝⁷ : Algebra S R
inst✝⁶ : IsScalarTower S R M
M₂ : Type u_4
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
inst✝³ : Module S M₂
inst✝² : SMulCommClass R S M₂
inst✝¹ : IsScalarTower S R M₂
inst✝ : Algebra.IsCentral S R
f g : M ≃ₗ[R] M₂
⊢ (∀ (a : End R M), a ∘ₗ ↑g.symm ∘ₗ ↑f = (↑g.symm ∘ₗ ↑f) ∘ₗ a) ↔ ∃ α, ⇑f = α • ⇑g
Module: Mathlib.RingTheory.Algebraic.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
A : Type u_3
inst✝⁶ : CommRing S
inst✝⁵ : NoZeroDivisors S
inst✝⁴ : Algebra R S
alg : Algebra.IsAlgebraic R S
inst✝³ : Ring A
inst✝² : Algebra R A
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
inj : Injective ⇑(algebraMap R A)
⊢ Injective ⇑(algebraMap S A)

Tactic:
refine (injective_iff_map_eq_zero _).mpr fun s eq ↦ of_not_not fun ne ↦ ?_
  have ⟨r, ne, dvd⟩ := (alg.1 s).exists_nonzero_dvd (mem_nonZeroDivisors_of_ne_zero ne)
  refine ne (inj <| map_zero (algebraMap R A) ▸ zero_dvd_iff.mp ?_)
  simp_rw [← eq, IsScalarTower.algebraMap_apply R S A, map_dvd (algebraMap S A) dvd]

Goal state after:
Module: Mathlib.Order.DirectedInverseSystem

Goal state before:
pp: case inl.inr
ι : Type u_6
F : ι → Type u_7
X : ι → Type u_8
i : ι
inst✝² : LinearOrder ι
f : ⦃i j : ι⦄ → i ≤ j → F j → F i
inst✝¹ : SuccOrder ι
equiv : (j : ↑(Iic i)) → F ↑j ≃ piLT X ↑j
e : F i⁺ ≃ F i × X i
hi : ¬IsMax i
inst✝ : InverseSystem f
H : ∀ (x : F i⁺), (e x).1 = f ⋯ x
nat : IsNatEquiv f equiv
k : ι
hk✝ : k ∈ Iic i⁺
lt_succ : ∀ {j : ι}, j ≤ i → j < i⁺
hj : i⁺ ∈ Iic i⁺
h : k ≤ i⁺
x : F i⁺
hk : k ≤ i
⊢ (piEquivSucc equiv e hi ⟨k, hk✝⟩) (f h x) = piLTProj h ((piEquivSucc equiv e hi ⟨i⁺, hj⟩) x)

Tactic:
funext l

Goal state after:
case inl.inr
ι : Type u_6
F : ι → Type u_7
X : ι → Type u_8
i : ι
inst✝² : LinearOrder ι
f : ⦃i j : ι⦄ → i ≤ j → F j → F i
inst✝¹ : SuccOrder ι
equiv : (j : ↑(Iic i)) → F ↑j ≃ piLT X ↑j
e : F i⁺ ≃ F i × X i
hi : ¬IsMax i
inst✝ : InverseSystem f
H : ∀ (x : F i⁺), (e x).1 = f ⋯ x
nat : IsNatEquiv f equiv
k : ι
hk✝ : k ∈ Iic i⁺
lt_succ : ∀ {j : ι}, j ≤ i → j < i⁺
hj : i⁺ ∈ Iic i⁺
h : k ≤ i⁺
x : F i⁺
hk : k ≤ i
l : ↑(Iio ↑⟨k, hk✝⟩)
⊢ (piEquivSucc equiv e hi ⟨k, hk✝⟩) (f h x) l = piLTProj h ((piEquivSucc equiv e hi ⟨i⁺, hj⟩) x) l
Module: Mathlib.Order.DirectedInverseSystem

Goal state before:
pp: case inl.inr
ι : Type u_6
F : ι → Type u_7
X : ι → Type u_8
inst✝¹ : LinearOrder ι
f : ⦃i j : ι⦄ → i ≤ j → F j → F i
inst✝ : InverseSystem f
j k : ι
h : k ≤ j
t : F j
equiv : (j_1 : ↑(Iio j)) → F ↑j_1 ≃ piLT X ↑j_1
nat : IsNatEquiv f equiv
equivLim : F j ≃ ↑(limit f j)
hi : IsSuccPrelimit j
H : ∀ (x : F j) (l : ↑(Iio j)), ↑(equivLim x) l = f ⋯ x
hj : j ∈ Iic j
hk✝ : k ∈ Iic j
hk : k < j
⊢ (piEquivLim nat equivLim hi ⟨k, hk✝⟩) (f h t) = piLTProj h ((piEquivLim nat equivLim hi ⟨j, hj⟩) t)

Tactic:
funext l

Goal state after:
case inl.inr
ι : Type u_6
F : ι → Type u_7
X : ι → Type u_8
inst✝¹ : LinearOrder ι
f : ⦃i j : ι⦄ → i ≤ j → F j → F i
inst✝ : InverseSystem f
j k : ι
h : k ≤ j
t : F j
equiv : (j_1 : ↑(Iio j)) → F ↑j_1 ≃ piLT X ↑j_1
nat : IsNatEquiv f equiv
equivLim : F j ≃ ↑(limit f j)
hi : IsSuccPrelimit j
H : ∀ (x : F j) (l : ↑(Iio j)), ↑(equivLim x) l = f ⋯ x
hj : j ∈ Iic j
hk✝ : k ∈ Iic j
hk : k < j
l : ↑(Iio ↑⟨k, hk✝⟩)
⊢ (piEquivLim nat equivLim hi ⟨k, hk✝⟩) (f h t) l = piLTProj h ((piEquivLim nat equivLim hi ⟨j, hj⟩) t) l
Module: Mathlib.Order.DirectedInverseSystem

Goal state before:
pp: ι✝ : Type u_1
inst✝³ : Preorder ι✝
F₁ : ι✝ → Type u_2
F₂ : ι✝ → Type u_3
F✝ : ι✝ → Type u_4
X✝ : ι✝ → Type u_5
f✝ : ⦃i j : ι✝⦄ → i ≤ j → F✝ j → F✝ i
i✝¹ j : ι✝
h : i✝¹ ≤ j
ι : Type u_6
F : ι → Type u_7
X : ι → Type u_8
i : ι
inst✝² : LinearOrder ι
f : ⦃i j : ι⦄ → i ≤ j → F j → F i
inst✝¹ : SuccOrder ι
equivSucc : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
s t : Set ι
inst✝ : WellFoundedLT ι
hi : IsSuccPrelimit i
e : (j : ↑(Iio i)) → PEquivOn f equivSucc (Iic ↑j)
i✝ : ι
hj : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
k : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
rw [piLTLim_symm_apply hi ⟨_, k.2.2⟩ (by exact k.2.1)]

Goal state after:
ι✝ : Type u_1
inst✝³ : Preorder ι✝
F₁ : ι✝ → Type u_2
F₂ : ι✝ → Type u_3
F✝ : ι✝ → Type u_4
X✝ : ι✝ → Type u_5
f✝ : ⦃i j : ι✝⦄ → i ≤ j → F✝ j → F✝ i
i✝¹ j : ι✝
h : i✝¹ ≤ j
ι : Type u_6
F : ι → Type u_7
X : ι → Type u_8
i : ι
inst✝² : LinearOrder ι
f : ⦃i j : ι⦄ → i ≤ j → F j → F i
inst✝¹ : SuccOrder ι
equivSucc : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
s t : Set ι
inst✝ : WellFoundedLT ι
hi : IsSuccPrelimit i
e : (j : ↑(Iio i)) → PEquivOn f equivSucc (Iic ↑j)
i✝ : ι
hj : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
k : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Module: Mathlib.RingTheory.MvPolynomial.Basic

Goal state before:
pp: case a
σ : Type u
R : Type v
inst✝ : CommSemiring R
s t : Set (σ →₀ ℕ)
⊢ ∀ x ∈ s, ∀ y ∈ t, x + y ∈ (fun x ↦ (monomial x) 1) ⁻¹' ↑(restrictSupport R s * restrictSupport R t)

Tactic:
intro x hx y hy

Goal state after:
case a
σ : Type u
R : Type v
inst✝ : CommSemiring R
s t : Set (σ →₀ ℕ)
x : σ →₀ ℕ
hx : x ∈ s
y : σ →₀ ℕ
hy : y ∈ t
⊢ x + y ∈ (fun x ↦ (monomial x) 1) ⁻¹' ↑(restrictSupport R s * restrictSupport R t)
Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: case e'_3
R : Type u_1
inst✝² : CommRing R
Q : Type u_5
inst✝¹ : AddCommGroup Q
inst✝ : Module R Q
I : Ideal R
⊢ comap (↑(TensorProduct.lid R Q)) (I • ⊤) = (rTensor Q (Submodule.subtype I)).range

Tactic:
comap_equiv_eq_map_symm,

Goal state after:
case e'_3
R : Type u_1
inst✝² : CommRing R
Q : Type u_5
inst✝¹ : AddCommGroup Q
inst✝ : Module R Q
I : Ideal R
⊢ Submodule.map (↑(TensorProduct.lid R Q).symm) (I • ⊤) = (rTensor Q (Submodule.subtype I)).range
Module: Mathlib.RingTheory.Nilpotent.Defs

Goal state before:
pp: case h
R : Type u_1
x y : R
inst✝ : Semiring R
h_comm : Commute x y
n : ℕ
hn : x ^ n = 0
⊢ 0 * y ^ n = 0

Tactic:
zero_mul

Goal state after:
case h
R : Type u_1
x y : R
inst✝ : Semiring R
h_comm : Commute x y
n : ℕ
hn : x ^ n = 0
⊢ 0 = 0