mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : Semiring A
inst✝¹ : HopfAlgebra R A
G : Type u_3
inst✝ : Group G
⊢ LinearMap.mul' R A[G] ∘ₗ LinearMap.lTensor A[G] (antipode R) ∘ₗ comul = Algebra.linearMap R A[G] ∘ₗ counit

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : HopfAlgebra R A
a : A
⊢ (antipode R) (C a) = C ((antipode R) a)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : HopfAlgebra R A
n : ℤ
⊢ (antipode R) (T n) = T (-n)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : HopfAlgebra R A
a : A
n : ℤ
⊢ (antipode R) (C a * T n) = C ((antipode R) a) * T (-n)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.IsAugmentation

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
A : Type u_2
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S : Subalgebra R A
I : Ideal A
⊢ IsAugmentation (↥S) I ↔ IsCompl (toSubmodule S) (Submodule.restrictScalars R I)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.HopfAlgebra.Quotient

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : HopfAlgebra R A
inst✝ : HopfAlgebraStruct R B
f : A →ₐc[R] B
hf : Function.Surjective ⇑f
hS : antipode R ∘ₗ f.toLinearMap = f.toLinearMap ∘ₗ antipode R
⊢ (toConv (antipode R) * toConv LinearMap.id).ofConv ∘ₗ f.toLinearMap =
    (toConv (f.toLinearMap ∘ₗ antipode R) * toConv f.toLinearMap).ofConv

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.HasGoingUp

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra.HasGoingUp R S
l : RelSeries {(a, b) | a < b}
q : PrimeSpectrum R
lt : (q, l.head) ∈ {(a, b) | a < b}
ih :
  ∀ (P : Ideal S) [inst : P.IsPrime] [lo : P.LiesOver l.head.asIdeal],
    ∃ L,
      L.length = l.length ∧
        RelSeries.head L = { asIdeal := P, isPrime := ⋯ } ∧
          List.map (PrimeSpectrum.comap (algebraMap R S)) (RelSeries.toList L) = l.toList
P : Ideal S
inst✝ : P.IsPrime
lo : P.LiesOver q.asIdeal
Q : Ideal S
PQlt : P < Q
hQ : Q.IsPrime
Qlo : Q.LiesOver l.head.asIdeal
L : LTSeries (PrimeSpectrum S)
len : L.length = l.length
head : RelSeries.head L = { asIdeal := Q, isPrime := ⋯ }
spec : List.map (PrimeSpectrum.comap (algebraMap R S)) (RelSeries.toList L) = l.toList
⊢ ({ asIdeal := P, isPrime := ⋯ }, RelSeries.head L) ∈ {(a, b) | a < b}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.HopfAlgebra.Quotient

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : HopfAlgebra R A
inst✝ : HopfAlgebraStruct R B
f : A →ₐc[R] B
hf : Function.Surjective ⇑f
hS : antipode R ∘ₗ f.toLinearMap = f.toLinearMap ∘ₗ antipode R
⊢ (toConv (f.toLinearMap ∘ₗ antipode R) * toConv f.toLinearMap).ofConv =
    (↑f).toLinearMap ∘ₗ (toConv (antipode R) * toConv LinearMap.id).ofConv

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.HasGoingUp

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra.HasGoingUp R S
l : RelSeries {(a, b) | a < b}
q : PrimeSpectrum R
lt : (q, l.head) ∈ {(a, b) | a < b}
ih :
  ∀ (P : Ideal S) [inst : P.IsPrime] [lo : P.LiesOver l.head.asIdeal],
    ∃ L,
      L.length = l.length ∧
        RelSeries.head L = { asIdeal := P, isPrime := ⋯ } ∧
          List.map (PrimeSpectrum.comap (algebraMap R S)) (RelSeries.toList L) = l.toList
P : Ideal S
inst✝ : P.IsPrime
lo : P.LiesOver q.asIdeal
Q : Ideal S
PQlt : P < Q
hQ : Q.IsPrime
Qlo : Q.LiesOver l.head.asIdeal
L : LTSeries (PrimeSpectrum S)
len : L.length = l.length
head : RelSeries.head L = { asIdeal := Q, isPrime := ⋯ }
spec : List.map (PrimeSpectrum.comap (algebraMap R S)) (RelSeries.toList L) = l.toList
⊢ (RelSeries.cons L { asIdeal := P, isPrime := ⋯ } ⋯).length = (l.cons q lt).length

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.HasGoingUp

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra.HasGoingUp R S
l : LTSeries (PrimeSpectrum R)
P : Ideal S
inst✝ : P.IsPrime
lo : P.LiesOver (RelSeries.head l).asIdeal
⊢ ∃ L,
    L.length = l.length ∧
      RelSeries.head L = { asIdeal := P, isPrime := ⋯ } ∧
        List.map (PrimeSpectrum.comap (algebraMap R S)) (RelSeries.toList L) = RelSeries.toList l

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.HopfAlgebra.Quotient

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : HopfAlgebra R A
inst✝ : HopfAlgebraStruct R B
f : A →ₐc[R] B
hf : Function.Surjective ⇑f
hS : antipode R ∘ₗ f.toLinearMap = f.toLinearMap ∘ₗ antipode R
⊢ (↑f).toLinearMap ∘ₗ (toConv (antipode R) * toConv LinearMap.id).ofConv = ofConv 1 ∘ₗ f.toLinearMap

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.HopfAlgebra.Quotient

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : HopfAlgebra R A
inst✝ : HopfAlgebraStruct R B
f : A →ₐc[R] B
hf : Function.Surjective ⇑f
hS : antipode R ∘ₗ f.toLinearMap = f.toLinearMap ∘ₗ antipode R
⊢ (toConv LinearMap.id * toConv (antipode R)).ofConv ∘ₗ f.toLinearMap =
    (toConv f.toLinearMap * toConv (f.toLinearMap ∘ₗ antipode R)).ofConv

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.HasGoingUp

Goal state before:
pp: R : Type u_3
S : Type u_4
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
⊢ HasGoingUp R S ↔ SpecializingMap (PrimeSpectrum.comap (algebraMap R S))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.HopfAlgebra.Quotient

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : HopfAlgebra R A
inst✝ : HopfAlgebraStruct R B
f : A →ₐc[R] B
hf : Function.Surjective ⇑f
hS : antipode R ∘ₗ f.toLinearMap = f.toLinearMap ∘ₗ antipode R
⊢ (toConv f.toLinearMap * toConv (f.toLinearMap ∘ₗ antipode R)).ofConv =
    (↑f).toLinearMap ∘ₗ (toConv LinearMap.id * toConv (antipode R)).ofConv

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.HasGoingUp

Goal state before:
pp: R : Type u_3
S : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : HasGoingUp R S
inst✝ : HasGoingUp S T
⊢ HasGoingUp R T

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
n : ℕ
ih : Polynomial.eval (c n) f ∈ I ^ (n + 1)
i : ℕ
h2i : 2 ≤ i
right✝ : i < ((taylor (c n)) f).natDegree + 2
⊢ n + 2 ≤ i * (n + 1)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.HopfAlgebra.Quotient

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : HopfAlgebra R A
inst✝ : HopfAlgebraStruct R B
f : A →ₐc[R] B
hf : Function.Surjective ⇑f
hS : antipode R ∘ₗ f.toLinearMap = f.toLinearMap ∘ₗ antipode R
⊢ (↑f).toLinearMap ∘ₗ (toConv LinearMap.id * toConv (antipode R)).ofConv = ofConv 1 ∘ₗ f.toLinearMap

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.HopfAlgebra.Quotient

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : HopfAlgebra R A
inst✝ : HopfAlgebraStruct R B
f : A →ₐc[R] B
hf : Function.Surjective ⇑f
hS : antipode R ∘ₗ f.toLinearMap = f.toLinearMap ∘ₗ antipode R
⊢ HopfAlgebra R B

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.HopfAlgebra.Quotient

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁴ : CommRing R
inst✝³ : Ring A
inst✝² : HopfAlgebraStruct R A
I : Ideal A
inst✝¹ : I.IsTwoSided
inst✝ : Ideal.IsHopfIdeal R I
⊢ antipode R ∘ₗ (mkₐ R I).toLinearMap = (mkₐ R I).toLinearMap ∘ₗ antipode R

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
⊢ ∀ (n : ℕ), Polynomial.eval (c n) f ∈ I ^ (n + 1)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
hfcI : ∀ (n : ℕ), Polynomial.eval (c n) f ∈ I ^ (n + 1)
⊢ ∀ (m n : ℕ), m ≤ n → c m ≡ c n [SMOD I ^ m • ⊤]

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
hfcI : ∀ (n : ℕ), Polynomial.eval (c n) f ∈ I ^ (n + 1)
aux : ∀ (m n : ℕ), m ≤ n → c m ≡ c n [SMOD I ^ m • ⊤]
a : R
ha : ∀ (n : ℕ), c n ≡ a [SMOD I ^ n • ⊤]
this : ∀ (n : ℕ), Polynomial.eval a f ≡ 0 [SMOD I ^ n • ⊤]
⊢ f.IsRoot a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsLocalRing R
I : Ideal R
hp✝ : maximalIdeal R ∈ I.minimalPrimes
hp : I.IsMinimalPrime (maximalIdeal R)
J : Ideal (R ⧸ I)
prime : J.IsPrime
this : (Ideal.comap (Ideal.Quotient.mk I) J).IsPrime
⊢ I ≤ RingHom.ker (algebraMap R (R ⧸ I))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsLocalRing R
I : Ideal R
hp : maximalIdeal R ∈ I.minimalPrimes
J : Ideal (R ⧸ I)
prime : J.IsPrime
⊢ (Ideal.comap (algebraMap (?m.34 J prime) (R ⧸ I)) J).IsMaximal

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
⊢ ∀ (f : R[X]),
    f.Monic →
      ∀ (a₀ : R),
        Polynomial.eval a₀ f ∈ I →
          IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f))) → ∃ a, f.IsRoot a ∧ a - a₀ ∈ I

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
f : R[X]
a b : R
ha : Polynomial.eval a f = 0
hb : Polynomial.eval b f = 0
h : ¬IsUnit (a - b)
h' : IsUnit (Polynomial.eval a (derivative f))
c : R
h✝ :
  c * (b - a) ^ 2 + Polynomial.eval a (derivative f) * (b - a) + Polynomial.eval a f = Polynomial.eval (a + (b - a)) f
⊢ (c * (b - a) + Polynomial.eval a (derivative f)) * (b - a) = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.KrullsHeightTheorem

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
inst✝² : IsNoetherianRing R
inst✝¹ : IsLocalRing R
I : Ideal R
inst✝ : Submodule.IsPrincipal I
hp : IsLocalRing.maximalIdeal R ∈ I.minimalPrimes
q : Ideal R
h₁ : q.IsPrime
h₂ : q < IsLocalRing.maximalIdeal R
this : q.height = 0
⊢ q.height < ↑1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.MonicSpan

Goal state before:
pp: k : Type u_2
inst✝ : Field k
I : Ideal k[X]
ne : I ≠ ⊥
⊢ ∃ f, f.Monic ∧ I = span {f}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
f : R[X]
a b : R
ha : Polynomial.eval a f = 0
hb : Polynomial.eval b f = 0
h : ¬IsUnit (a - b)
h' : IsUnit (Polynomial.eval a (derivative f))
c : R
h✝ :
  c * (b - a) ^ 2 + Polynomial.eval a (derivative f) * (b - a) + Polynomial.eval a f = Polynomial.eval (a + (b - a)) f
hc : (c * (b - a) + Polynomial.eval a (derivative f)) * (b - a) = 0
this : c * (b - a) + Polynomial.eval a (derivative f) ∉ maximalIdeal R
⊢ a = b

Tactic:
rw [notMem_maximalIdeal, isUnit_iff_exists] at this
    grind

Goal state after:
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
f : R[X]
a b : R
ha : Polynomial.eval a f = 0
hb : Polynomial.eval b f = 0
h : ¬IsUnit (a - b)
h' : IsUnit (Polynomial.eval a (derivative f))
c : R
h✝ :
  c * (b - a) ^ 2 + Polynomial.eval a (derivative f) * (b - a) + Polynomial.eval a f = Polynomial.eval (a + (b - a)) f
hc : (c * (b - a) + Polynomial.eval a (derivative f)) * (b - a) = 0
this : c * (b - a) + Polynomial.eval a (derivative f) ∉ maximalIdeal R
⊢ a = b

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.MonicSpan

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
p : Ideal R[X]
ism : (comap C p).IsMaximal
ne : p ≠ Ideal.map C (comap C p)
q : Ideal R := comap C p
this : Field (R ⧸ q) := Quotient.field q
⊢ Ideal.map (mapRingHom (Ideal.Quotient.mk q)) p ≠ ⊥

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
f : R[X]
a b : R
ha : Polynomial.eval a f = 0
hb : Polynomial.eval b f = 0
h : ¬IsUnit (a - b)
h' : IsUnit (Polynomial.eval a (derivative f))
⊢ a = b

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.UFD

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : WfDvdMonoid R
x : R
hx : Prime x
p : Ideal R
inst✝³ : p.IsPrime
hxp : x ∉ p
S : Type u_2
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsLocalization.Away x S
hp : Submodule.IsPrincipal (map (algebraMap R S) p)
this : Disjoint ↑(Submonoid.powers x) ↑p
hpbot : ¬p = ⊥
hi : Function.Injective ⇑(algebraMap R S)
⊢ map (algebraMap R S) p ≠ ⊥

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.UFD

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : WfDvdMonoid R
x : R
hx : Prime x
p : Ideal R
inst✝³ : p.IsPrime
hxp : x ∉ p
S : Type u_2
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsLocalization.Away x S
this : Disjoint ↑(Submonoid.powers x) ↑p
hpbot : ¬p = ⊥
hi : Function.Injective ⇑(algebraMap R S)
hpb : map (algebraMap R S) p ≠ ⊥
g : S
hg : map (algebraMap R S) p = S ∙ g
hg0 : g = 0
⊢ map (algebraMap R S) p = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.UFD

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : WfDvdMonoid R
x : R
hx : Prime x
p : Ideal R
inst✝³ : p.IsPrime
hxp : x ∉ p
S : Type u_2
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsLocalization.Away x S
this : Disjoint ↑(Submonoid.powers x) ↑p
hpbot : ¬p = ⊥
hi : Function.Injective ⇑(algebraMap R S)
hpb : map (algebraMap R S) p ≠ ⊥
g : S
hg : map (algebraMap R S) p = S ∙ g
hg0 : g ≠ 0
a : R
n : ℤ
hxa : ¬x ∣ a
hag : selfZPow x S n * (algebraMap R S) a = g
hu : IsUnit (selfZPow x S n)
⊢ ∀ (y : R), x * y ∈ p → y ∈ p

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.UFD

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : WfDvdMonoid R
x : R
hx : Prime x
p : Ideal R
inst✝³ : p.IsPrime
hxp : x ∉ p
S : Type u_2
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsLocalization.Away x S
hp : Submodule.IsPrincipal (map (algebraMap R S) p)
⊢ Submodule.IsPrincipal p

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.UFD

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : UniqueFactorizationMonoid R
p : Ideal R
inst✝ : p.IsPrime
hph : p.height = 1
⊢ Submodule.IsPrincipal p

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.UFD

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsNoetherianRing R
h : ∀ (p : Ideal R) [p.IsPrime], p.height = 1 → Submodule.IsPrincipal p
I : Ideal R
hIn : I ≠ ⊥
a✝ : I.IsPrime
x : R
hxI : x ∈ I
hx0 : x ≠ 0
p : Ideal R
hpmin : p ∈ (Ideal.span {x}).minimalPrimes
hpl : p ≤ I
this : p.IsPrime
hpn : p ≠ ⊥
⊢ 1 ≤ p.height

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.UFD

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsNoetherianRing R
h : ∀ (p : Ideal R) [p.IsPrime], p.height = 1 → Submodule.IsPrincipal p
⊢ UniqueFactorizationMonoid R

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.UFD

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsNoetherianRing R
x : R
hx : Prime x
S : Type u_2
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsLocalization.Away x S
this : IsDomain S
x✝ : UniqueFactorizationMonoid S
p : Ideal R
hp : p.IsPrime
h1 : p.height = 1
hxp : x ∉ p
⊢ ?m.77

Tactic:
rwa [← Ideal.disjoint_powers_iff_notMem_of_isPrime x] at hxp

Goal state after:
Module: Mathlib.RingTheory.Ideal.UFD

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsNoetherianRing R
x : R
hx : Prime x
S : Type u_2
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsLocalization.Away x S
this : IsDomain S
x✝ : UniqueFactorizationMonoid S
p : Ideal R
hp : p.IsPrime
h1 : p.height = 1
hxp : x ∉ p
⊢ ?m.77

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.MonicSpan

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
p : Ideal R[X]
ism : (comap C p).IsMaximal
ne : p ≠ Ideal.map C (comap C p)
⊢ ∃ f, f.Monic ∧ p = Ideal.map C (comap C p) ⊔ span {f}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Ideal.UFD

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsNoetherianRing R
x : R
hx : Prime x
S : Type u_2
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsLocalization.Away x S
⊢ UniqueFactorizationMonoid R ↔ UniqueFactorizationMonoid S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Basic

Goal state before:
pp: A : Type u_1
inst✝ : Ring A
F : IdealFilter A
L K : Ideal A
h : F.IsTorsionQuot (L ⊓ K) K
k : A
hk : k ∈ K
I : Ideal A
hI : I ∈ F
hI_le : I ≤ Submodule.colon (L ⊓ K) {k}
hcol : Submodule.colon (L ⊓ K) {k} = Submodule.colon L {k}
⊢ I ≤ Submodule.colon L {k}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Basic

Goal state before:
pp: A : Type u_1
inst✝ : Ring A
F : IdealFilter A
L K : Ideal A
h : F.IsTorsionQuot L K
k : A
hk : k ∈ K
I : Ideal A
hI : I ∈ F
hI_le : I ≤ Submodule.colon L {k}
hcol : Submodule.colon (L ⊓ K) {k} = Submodule.colon L {k}
⊢ I ≤ Submodule.colon (L ⊓ K) {k}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Basic

Goal state before:
pp: A : Type u_1
inst✝ : Ring A
F : IdealFilter A
L K : Ideal A
⊢ F.IsTorsionQuot (L ⊓ K) K ↔ F.IsTorsionQuot L K

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Basic

Goal state before:
pp: A : Type u_1
inst✝ : Ring A
F : IdealFilter A
I : Ideal A
x : A
hx : x ∈ I
J : Ideal A
hJ : J ∈ ↑F
⊢ ⊤ = Submodule.colon I {x}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Basic

Goal state before:
pp: A : Type u_1
inst✝ : Ring A
F : IdealFilter A
I : Ideal A
⊢ F.IsTorsionQuot I I

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Topology

Goal state before:
pp: A : Type u_1
inst✝ : Ring A
F : IdealFilter A
⊢ ∀ {x y : Set A}, x ∈ {x | ∃ I ∈ F, ↑I = x} → y ∈ {x | ∃ I ∈ F, ↑I = x} → ∃ z ∈ {x | ∃ I ∈ F, ↑I = x}, z ⊆ x ∩ y

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Topology

Goal state before:
pp: A : Type u_1
inst✝ : Ring A
F : IdealFilter A
⊢ ∀ {U : Set A}, U ∈ {x | ∃ I ∈ F, ↑I = x} → 0 ∈ U

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Topology

Goal state before:
pp: A : Type u_1
inst✝ : Ring A
F : IdealFilter A
⊢ ∀ {U : Set A}, U ∈ {x | ∃ I ∈ F, ↑I = x} → ∃ V ∈ {x | ∃ I ∈ F, ↑I = x}, V + V ⊆ U

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Topology

Goal state before:
pp: A : Type u_1
inst✝ : Ring A
F : IdealFilter A
⊢ ∀ {U : Set A}, U ∈ {x | ∃ I ∈ F, ↑I = x} → ∃ V ∈ {x | ∃ I ∈ F, ↑I = x}, V ⊆ (fun x ↦ -x) ⁻¹' U

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Topology

Goal state before:
pp: A : Type u_1
inst✝ : Ring A
F : IdealFilter A
⊢ ∀ (x₀ : A) {U : Set A}, U ∈ {x | ∃ I ∈ F, ↑I = x} → ∃ V ∈ {x | ∃ I ∈ F, ↑I = x}, V ⊆ (fun x ↦ x₀ + x + -x₀) ⁻¹' U

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Topology

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
F : IdealFilter A
inst✝ : F.IsUniform
⊢ ∀ {U : Set A}, U ∈ F.addGroupFilterBasis.sets → ∃ V ∈ F.addGroupFilterBasis.sets, V * V ⊆ U

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Topology

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
F : IdealFilter A
inst✝ : F.IsUniform
⊢ ∀ (x₀ : A) {U : Set A}, U ∈ F.addGroupFilterBasis.sets → ∃ V ∈ F.addGroupFilterBasis.sets, V ⊆ (fun x ↦ x₀ * x) ⁻¹' U

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Topology

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
F : IdealFilter A
inst✝ : F.IsUniform
⊢ ∀ (x₀ : A) {U : Set A}, U ∈ F.addGroupFilterBasis.sets → ∃ V ∈ F.addGroupFilterBasis.sets, V ⊆ (fun x ↦ x * x₀) ⁻¹' U

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Basic

Goal state before:
pp: A : Type u_1
inst✝ : Ring A
F : IdealFilter A
I J K : Ideal A
hI : F.IsTorsionQuot I K
hJ : F.IsTorsionQuot J K
⊢ F.IsTorsionQuot (I ⊓ J) K

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Topology

Goal state before:
pp: A : Type u_1
inst✝ : Ring A
F : IdealFilter A
x✝ : ∃ B, B.sets = {x | ∃ I ∈ F, ↑I = x}
B : RingFilterBasis A
hB : B.sets = {x | ∃ I ∈ F, ↑I = x}
I : Ideal A
hI : I ∈ F
a : A
⊢ ↑I ∈ {x | ∃ I ∈ F, ↑I = x}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Basic

Goal state before:
pp: A : Type u_1
inst✝ : Ring A
F G : IdealFilter A
⊢ Order.IsPFilter {L | ∃ K ∈ G, F.IsTorsionQuot L K}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Topology

Goal state before:
pp: A : Type u_1
inst✝ : Ring A
F : IdealFilter A
x✝ : ∃ B, B.sets = {x | ∃ I ∈ F, ↑I = x}
B : RingFilterBasis A
hB : B.sets = {x | ∃ I ∈ F, ↑I = x}
I : Ideal A
hI : I ∈ F
a : A
J : Ideal A
hJ : J ∈ F
hbasis : ↑J ∈ B
hsub : ↑J ⊆ (fun x ↦ x * a) ⁻¹' ↑I
x : A
hx : x ∈ J
⊢ x ∈ Submodule.colon I {a}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.IdealFilter.Topology

Goal state before:
pp: A : Type u_1
inst✝ : Ring A
F : IdealFilter A
⊢ F.IsUniform ↔ ∃ B, B.sets = {x | ∃ I ∈ F, ↑I = x}

Tactic:
by

Goal state after: