mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.FieldTheory.Galois.IsGaloisGroup

Goal state before:
pp: G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : Finite G
A : Type u_5
B : Type u_6
C : Type u_7
H : Subgroup G
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : CommRing C
inst✝⁵ : IsDomain C
inst✝⁴ : Algebra A C
inst✝³ : FaithfulSMul A C
inst✝² : MulSemiringAction G C
hGAC : IsGaloisGroup G A C
inst✝¹ : Algebra B C
inst✝ : FaithfulSMul B C
hH : IsGaloisGroup (↥H) B C
this✝² : IsDomain B
this✝¹ : IsDomain A
K : Type u_5 := FractionRing A
L : Type u_7 := FractionRing C
this✝ : MulSemiringAction G L := IsFractionRing.mulSemiringAction G C L
this : IsGaloisGroup (↥H) (FractionRing B) L
g : G
⊢ g ∈ fixingSubgroup G (Set.range ⇑(algebraMap B C)) ↔ g ∈ fixingSubgroup G (Set.range ⇑(algebraMap (FractionRing B) L))

Tactic:
simp only [mem_fixingSubgroup_iff, Set.mem_range]

Goal state after:
G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : Finite G
A : Type u_5
B : Type u_6
C : Type u_7
H : Subgroup G
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : CommRing C
inst✝⁵ : IsDomain C
inst✝⁴ : Algebra A C
inst✝³ : FaithfulSMul A C
inst✝² : MulSemiringAction G C
hGAC : IsGaloisGroup G A C
inst✝¹ : Algebra B C
inst✝ : FaithfulSMul B C
hH : IsGaloisGroup (↥H) B C
this✝² : IsDomain B
this✝¹ : IsDomain A
K : Type u_5 := FractionRing A
L : Type u_7 := FractionRing C
this✝ : MulSemiringAction G L := IsFractionRing.mulSemiringAction G C L
this : IsGaloisGroup (↥H) (FractionRing B) L
g : G
⊢ (∀ (y : C), (∃ y_1, (algebraMap B C) y_1 = y) → g • y = y) ↔
    ∀ (y : L), (∃ y_1, (algebraMap (FractionRing B) L) y_1 = y) → g • y = y
Module: Mathlib.FieldTheory.Galois.IsGaloisGroup

Goal state before:
pp: G : Type u_1
K : Type u_3
L : Type u_4
inst✝¹¹ : Group G
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
inst✝⁷ : MulSemiringAction G L
H : Subgroup G
hGKL : IsGaloisGroup G K L
E : Type u_5
inst✝⁶ : Field E
inst✝⁵ : Algebra K E
inst✝⁴ : Algebra E L
inst✝³ : IsScalarTower K E L
inst✝² : Finite G
inst✝¹ : IsGaloisGroup (↥H) E L
inst✝ : IsGalois K E
F : IntermediateField K L := (IsScalarTower.toAlgHom K E L).fieldRange
this✝¹ : IsGalois K ↥F
hFL : IsGaloisGroup (↥H) (↥F) L
this✝ : IsGalois K L
this : Finite Gal(L/K)
⊢ (fixingSubgroup G ↑F).Normal

Tactic:
← mulEquivCongr_mapSubgroup_fixingSubgroup Gal(L/K) G K,

Goal state after:
G : Type u_1
K : Type u_3
L : Type u_4
inst✝¹¹ : Group G
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
inst✝⁷ : MulSemiringAction G L
H : Subgroup G
hGKL : IsGaloisGroup G K L
E : Type u_5
inst✝⁶ : Field E
inst✝⁵ : Algebra K E
inst✝⁴ : Algebra E L
inst✝³ : IsScalarTower K E L
inst✝² : Finite G
inst✝¹ : IsGaloisGroup (↥H) E L
inst✝ : IsGalois K E
F : IntermediateField K L := (IsScalarTower.toAlgHom K E L).fieldRange
this✝¹ : IsGalois K ↥F
hFL : IsGaloisGroup (↥H) (↥F) L
this✝ : IsGalois K L
this : Finite Gal(L/K)
⊢ (Subgroup.map (↑(mulEquivCongr Gal(L/K) G K L)) (fixingSubgroup Gal(L/K) ↑F)).Normal
Module: Mathlib.FieldTheory.Galois.IsGaloisGroup

Goal state before:
pp: G : Type u_1
G' : Type u_2
inst✝¹⁷ : Group G
inst✝¹⁶ : Group G'
A : Type u_5
B : Type u_6
C : Type u_7
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : CommRing C
inst✝¹² : IsDomain C
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Algebra A C
inst✝⁹ : Algebra B C
inst✝⁸ : FaithfulSMul A B
inst✝⁷ : FaithfulSMul B C
inst✝⁶ : IsScalarTower A B C
inst✝⁵ : Finite G
inst✝⁴ : Finite G'
inst✝³ : MulSemiringAction G C
inst✝² : IsGaloisGroup G A C
inst✝¹ : MulSemiringAction G' B
inst✝ : IsGaloisGroup G' A B
⊢ Function.Surjective ⇑(restrictHom G G' A B C)

Tactic:
by
  simpa [restrictHom] using QuotientGroup.mk_surjective

Goal state after:
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: K : Type u_1
inst✝³ : Field K
inst✝² : Fintype K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
n : ℕ
hcard : q = p ^ n
x : K
⊢ (frobenius K p ^ n) x = 1 x

Tactic:
conv_rhs => rw [RingHom.one_def, RingHom.id_apply, ← pow_card x, hcard]

Goal state after:
K : Type u_1
inst✝³ : Field K
inst✝² : Fintype K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
n : ℕ
hcard : q = p ^ n
x : K
⊢ (frobenius K p ^ n) x = x ^ p ^ n
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CommMonoidWithZero β
f : α → β
a b : α
h0 : f 0 = 0
h1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y
hpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i
hcp : ∀ {x y : α}, IsRelPrime x y → f (x * y) = f x * f y
this✝² : DecidableEq α := Classical.decEq α
ha0 : ¬a = 0
hb0 : ¬b = 0
hf1 : ¬f 1 = 0
this✝¹ : Nontrivial α
this✝ : StrongNormalizationMonoid α := UniqueFactorizationMonoid.strongNormalizationMonoid
this :
  f
      (∏ p ∈ (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset,
        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =
    f
        (∏ p ∈ (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset,
          p ^ Multiset.count p (normalizedFactors a)) *
      f
        (∏ p ∈ (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset,
          p ^ Multiset.count p (normalizedFactors b))
ua : αˣ
a_eq : (normalizedFactors a).prod * ↑ua = a
ub : αˣ
b_eq : (normalizedFactors b).prod * ↑ub = b
⊢ f ((normalizedFactors a).prod * (normalizedFactors b).prod) * f ↑ub * f ↑ua =
    f (normalizedFactors a).prod * f ↑ua * (f (normalizedFactors b).prod * f ↑ub)

Tactic:
mul_right_comm _ (f ua),

Goal state after:
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
β : Type u_3
inst✝ : CommMonoidWithZero β
f : α → β
a b : α
h0 : f 0 = 0
h1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y
hpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i
hcp : ∀ {x y : α}, IsRelPrime x y → f (x * y) = f x * f y
this✝² : DecidableEq α := Classical.decEq α
ha0 : ¬a = 0
hb0 : ¬b = 0
hf1 : ¬f 1 = 0
this✝¹ : Nontrivial α
this✝ : StrongNormalizationMonoid α := UniqueFactorizationMonoid.strongNormalizationMonoid
this :
  f
      (∏ p ∈ (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset,
        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =
    f
        (∏ p ∈ (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset,
          p ^ Multiset.count p (normalizedFactors a)) *
      f
        (∏ p ∈ (normalizedFactors a).toFinset ∪ (normalizedFactors b).toFinset,
          p ^ Multiset.count p (normalizedFactors b))
ua : αˣ
a_eq : (normalizedFactors a).prod * ↑ua = a
ub : αˣ
b_eq : (normalizedFactors b).prod * ↑ub = b
⊢ f ((normalizedFactors a).prod * (normalizedFactors b).prod) * f ↑ub * f ↑ua =
    f (normalizedFactors a).prod * (f (normalizedFactors b).prod * f ↑ub) * f ↑ua
Module: Mathlib.RingTheory.Artinian.Ring

Goal state before:
pp: case h
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsArtinianRing R
S : Submonoid R
L : Type u_2
inst✝² : CommSemiring L
inst✝¹ : Algebra R L
inst✝ : IsLocalization S L
r₁ : R
s : ↥S
n : ℕ
r : R
hr : ↑s ^ n.succ • r = ↑s ^ n • 1
⊢ IsLocalization.mk' L 1 s = (algebraMap R L) r

Tactic:
rw [smul_eq_mul, smul_eq_mul, pow_succ, mul_assoc] at hr

Goal state after:
case h
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsArtinianRing R
S : Submonoid R
L : Type u_2
inst✝² : CommSemiring L
inst✝¹ : Algebra R L
inst✝ : IsLocalization S L
r₁ : R
s : ↥S
n : ℕ
r : R
hr : ↑s ^ n * (↑s * r) = ↑s ^ n * 1
⊢ IsLocalization.mk' L 1 s = (algebraMap R L) r
Module: Mathlib.RingTheory.Finiteness.NilpotentKer

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : CommRing T
inst✝² : Algebra R S
inst✝¹ : Algebra R T
inst✝ : Module.Finite R T
f : S →ₐ[R] T
hf₁ : Function.Surjective ⇑f
I : Ideal S
hI : RingHom.ker f = I
hf₂ : I ≤ nilradical S
hf₃ : I.FG
this : Module.Finite R (S ⧸ I)
⊢ Module.Finite R S

Tactic:
suffices ∀ i, Module.Finite R (S ⧸ I ^ i) by
    obtain ⟨n, hn : _ = ⊥⟩ := hf₃.isNilpotent_iff_le_nilradical.mpr hf₂
    let e : (S ⧸ I ^ n) ≃ₐ[R] S := hn ▸ (AlgEquiv.quotientBot R S)
    exact .of_surjective e.toLinearMap e.surjective

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : CommRing T
inst✝² : Algebra R S
inst✝¹ : Algebra R T
inst✝ : Module.Finite R T
f : S →ₐ[R] T
hf₁ : Function.Surjective ⇑f
I : Ideal S
hI : RingHom.ker f = I
hf₂ : I ≤ nilradical S
hf₃ : I.FG
this : Module.Finite R (S ⧸ I)
⊢ ∀ (i : ℕ), Module.Finite R (S ⧸ I ^ i)
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: case succ.refine_1
S : Type u_1
inst✝¹ : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst✝ : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot (P ^ i) = cardQuot P ^ i
this : P ^ (i + 1) < P ^ i
a : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
f g : (c : S) → c ∈ P ^ i → S
hg : ∀ (c : S) (hc : c ∈ P ^ i), g c hc ∈ P ^ (i + 1)
hf : ∀ (c : S) (hc : c ∈ P ^ i), a * f c hc + g c hc = c
k : (c' : S ⧸ P ^ i.succ) → c' ∈ map (mkQ (P ^ i.succ)) (P ^ i) → S
hk_mem : ∀ (c' : S ⧸ P ^ i.succ) (hc' : c' ∈ map (mkQ (P ^ i.succ)) (P ^ i)), k c' hc' ∈ P ^ i
hk_eq : ∀ (c' : S ⧸ P ^ i.succ) (hc' : c' ∈ map (mkQ (P ^ i.succ)) (P ^ i)), (mkQ (P ^ i.succ)) (k c' hc') = c'
c₁' : S ⧸ P ^ i.succ
hc₁' : c₁' ∈ map (mkQ (P ^ i.succ)) (P ^ i)
c₂' : S ⧸ P ^ i.succ
hc₂' : c₂' ∈ map (mkQ (P ^ i.succ)) (P ^ i)
h : (fun c' ↦ Quotient.mk'' (f (k ↑c' ⋯) ⋯)) ⟨c₁', hc₁'⟩ = (fun c' ↦ Quotient.mk'' (f (k ↑c' ⋯) ⋯)) ⟨c₂', hc₂'⟩
⊢ a * f (k c₁' hc₁') ⋯ + g (k c₁' hc₁') ⋯ - (a * f (k c₂' hc₂') ⋯ + g (k c₂' hc₂') ⋯) ∈ P ^ i.succ

Tactic:
refine Ideal.mul_add_mem_pow_succ_inj _ _ _ _ _ _ a_mem (hg _ _) (hg _ _) ?_

Goal state after:
case succ.refine_1
S : Type u_1
inst✝¹ : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst✝ : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot (P ^ i) = cardQuot P ^ i
this : P ^ (i + 1) < P ^ i
a : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
f g : (c : S) → c ∈ P ^ i → S
hg : ∀ (c : S) (hc : c ∈ P ^ i), g c hc ∈ P ^ (i + 1)
hf : ∀ (c : S) (hc : c ∈ P ^ i), a * f c hc + g c hc = c
k : (c' : S ⧸ P ^ i.succ) → c' ∈ map (mkQ (P ^ i.succ)) (P ^ i) → S
hk_mem : ∀ (c' : S ⧸ P ^ i.succ) (hc' : c' ∈ map (mkQ (P ^ i.succ)) (P ^ i)), k c' hc' ∈ P ^ i
hk_eq : ∀ (c' : S ⧸ P ^ i.succ) (hc' : c' ∈ map (mkQ (P ^ i.succ)) (P ^ i)), (mkQ (P ^ i.succ)) (k c' hc') = c'
c₁' : S ⧸ P ^ i.succ
hc₁' : c₁' ∈ map (mkQ (P ^ i.succ)) (P ^ i)
c₂' : S ⧸ P ^ i.succ
hc₂' : c₂' ∈ map (mkQ (P ^ i.succ)) (P ^ i)
h : (fun c' ↦ Quotient.mk'' (f (k ↑c' ⋯) ⋯)) ⟨c₁', hc₁'⟩ = (fun c' ↦ Quotient.mk'' (f (k ↑c' ⋯) ⋯)) ⟨c₂', hc₂'⟩
⊢ f (k c₁' hc₁') ⋯ - f (k c₂' hc₂') ⋯ ∈ P
Module: Mathlib.RingTheory.Spectrum.Prime.Jacobson

Goal state before:
pp: case mp
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsJacobsonRing R
p : PrimeSpectrum R
tfae_1_to_2 : IsOpen {p} → IsClopen {p}
tfae_2_to_3 : IsClopen {p} → IsClosed {p} ∧ StableUnderGeneralization {p}
h₁ : IsMax p
h₂ : IsMin p
i✝ : PrimeSpectrum R
a✝ : IsMin i✝
⊢ i✝ ≤ p → i✝ = p

Tactic:
exact h₂.eq_of_le

Goal state after:
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : JacobsonSpace X
h : (closedPoints X).Finite
this : closedPoints X = Set.univ
inst : Finite X
s : Set X
x : X
x✝ : x ∈ sᶜ
⊢ IsClosed[inst✝¹] {x}

Tactic:
rw [← mem_closedPoints_iff, this]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : JacobsonSpace X
h : (closedPoints X).Finite
this : closedPoints X = Set.univ
inst : Finite X
s : Set X
x : X
x✝ : x ∈ sᶜ
⊢ x ∈ Set.univ
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
x : S
hx : Prime ((Algebra.norm ℤ) x)
y : ℤ
⊢ (Algebra.norm ℤ) x ∣ y ↔ y ∈ comap (algebraMap ℤ S) (span {x})

Tactic:
← Ideal.span_singleton_absNorm,

Goal state after:
S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
x : S
hx : Prime ((Algebra.norm ℤ) x)
y : ℤ
⊢ (Algebra.norm ℤ) x ∣ y ↔ y ∈ span {↑(absNorm (span {x}))}

S : Type u_1
inst✝³ : CommRing S
inst✝² : IsDedekindDomain S
inst✝¹ : Free ℤ S
inst✝ : Module.Finite ℤ S
x : S
hx : Prime ((Algebra.norm ℤ) x)
y : ℤ
⊢ Nat.Prime (absNorm (span {x}))
Module: Mathlib.RingTheory.RamificationInertia.Inertia

Goal state before:
pp: case pos
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
q : Ideal S
r : Ideal T
inst✝ : r.LiesOver q
hr : r.IsPrime
⊢ r.inertiaDeg R = q.inertiaDeg R * r.inertiaDeg S

Tactic:
have : q.IsPrime := isPrime_of_liesOver r q

Goal state after:
case pos
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
q : Ideal S
r : Ideal T
inst✝ : r.LiesOver q
hr : r.IsPrime
this : q.IsPrime
⊢ r.inertiaDeg R = q.inertiaDeg R * r.inertiaDeg S
Module: Mathlib.RingTheory.RamificationInertia.Inertia

Goal state before:
pp: case neg
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
q : Ideal S
r : Ideal T
inst✝ : r.LiesOver q
hr : ¬r.IsPrime
⊢ r.inertiaDeg R = q.inertiaDeg R * r.inertiaDeg S

Tactic:
rw [inertiaDeg_of_not_isPrime r R hr, inertiaDeg_of_not_isPrime r S hr, mul_zero]

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Instances

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹² : CommRing R
inst✝¹¹ : CommRing A
inst✝¹⁰ : CommRing B
inst✝⁹ : Algebra R A
inst✝⁸ : Algebra A B
inst✝⁷ : Algebra R B
inst✝⁶ : IsScalarTower R A B
p : Ideal A
q : Ideal B
inst✝⁵ : q.LiesOver p
inst✝⁴ : p.IsMaximal
inst✝³ : q.IsMaximal
inst✝² : Algebra (Localization.AtPrime p) (Localization.AtPrime q)
inst✝¹ : Localization.AtPrime.IsLiesOverAlgebra p q
inst✝ : Algebra.IsSeparable (A ⧸ p) (B ⧸ q)
⊢ Algebra.IsSeparable p.ResidueField q.ResidueField

Tactic:
refine Algebra.IsSeparable.of_equiv_equiv
    (.ofBijective _ p.bijective_algebraMap_quotient_residueField)
    (.ofBijective _ q.bijective_algebraMap_quotient_residueField) ?_

Goal state after:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹² : CommRing R
inst✝¹¹ : CommRing A
inst✝¹⁰ : CommRing B
inst✝⁹ : Algebra R A
inst✝⁸ : Algebra A B
inst✝⁷ : Algebra R B
inst✝⁶ : IsScalarTower R A B
p : Ideal A
q : Ideal B
inst✝⁵ : q.LiesOver p
inst✝⁴ : p.IsMaximal
inst✝³ : q.IsMaximal
inst✝² : Algebra (Localization.AtPrime p) (Localization.AtPrime q)
inst✝¹ : Localization.AtPrime.IsLiesOverAlgebra p q
inst✝ : Algebra.IsSeparable (A ⧸ p) (B ⧸ q)
⊢ (algebraMap p.ResidueField q.ResidueField).comp ↑(RingEquiv.ofBijective (algebraMap (A ⧸ p) p.ResidueField) ⋯) =
    (↑(RingEquiv.ofBijective (algebraMap (B ⧸ q) q.ResidueField) ⋯)).comp (algebraMap (A ⧸ p) (B ⧸ q))
Module: Mathlib.RingTheory.Unramified.Basic

Goal state before:
pp: R : Type v
inst✝⁵ : CommRing R
A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
B : Type w
inst✝² : CommRing B
inst✝¹ : Algebra R B
I : Ideal B
inst✝ : FormallyUnramified R A
hI : ⨅ i, I ^ i = ⊥
g₁ g₂ : A →ₐ[R] B
H : ∀ (x : A), (Ideal.Quotient.mk I) (g₁ x) = (Ideal.Quotient.mk I) (g₂ x)
this : ∀ (i : ℕ), (Ideal.Quotient.mkₐ R (I ^ i)).comp g₁ = (Ideal.Quotient.mkₐ R (I ^ i)).comp g₂
x : A
⊢ g₁ x - g₂ x = 0

Tactic:
← Ideal.mem_bot,

Goal state after:
R : Type v
inst✝⁵ : CommRing R
A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
B : Type w
inst✝² : CommRing B
inst✝¹ : Algebra R B
I : Ideal B
inst✝ : FormallyUnramified R A
hI : ⨅ i, I ^ i = ⊥
g₁ g₂ : A →ₐ[R] B
H : ∀ (x : A), (Ideal.Quotient.mk I) (g₁ x) = (Ideal.Quotient.mk I) (g₂ x)
this : ∀ (i : ℕ), (Ideal.Quotient.mkₐ R (I ^ i)).comp g₁ = (Ideal.Quotient.mkₐ R (I ^ i)).comp g₂
x : A
⊢ g₁ x - g₂ x ∈ ⊥
Module: Mathlib.RingTheory.QuasiFinite.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : QuasiFinite R S
I : Ideal R
⊢ (I.primesOver S).Finite

Tactic:
by_cases h : I.IsPrime

Goal state after:
case pos
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : QuasiFinite R S
I : Ideal R
h : I.IsPrime
⊢ (I.primesOver S).Finite

case neg
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : QuasiFinite R S
I : Ideal R
h : ¬I.IsPrime
⊢ (I.primesOver S).Finite
Module: Mathlib.RingTheory.Unramified.Finite

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FormallyUnramified R S
inst✝¹ : EssFiniteType R S
inst✝ : Free R S
I : Type u_2 := Free.ChooseBasisIndex R S
b : Basis I R S := Free.chooseBasis R S
b' : Basis I S (S ⊗[R] S) := ((Basis.singleton PUnit.{1} S).tensorProduct b).reindex (Equiv.punitProd I)
| (Finsupp.linearCombination S ⇑b') (b'.repr (elem R S))

Tactic:
Finsupp.linearCombination_apply

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FormallyUnramified R S
inst✝¹ : EssFiniteType R S
inst✝ : Free R S
I : Type u_2 := Free.ChooseBasisIndex R S
b : Basis I R S := Free.chooseBasis R S
b' : Basis I S (S ⊗[R] S) := ((Basis.singleton PUnit.{1} S).tensorProduct b).reindex (Equiv.punitProd I)
| (b'.repr (elem R S)).sum fun i a ↦ a • b' i
Module: Mathlib.RingTheory.Smooth.Basic

Goal state before:
pp: R : Type u
A : Type v
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
B : Type u_1
P : Type u_2
C : Type u_3
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R B
inst✝⁴ : CommRing C
inst✝³ : Algebra R C
inst✝² : CommRing P
inst✝¹ : Algebra R P
P₁ : Extension R A
P₂ : Extension R A
inst✝ : FormallySmooth R P₁.Ring
lift : P₁.Ring →ₐ[R] P₂.infinitesimal.Ring :=
  FormallySmooth.liftOfSurjective (IsScalarTower.toAlgHom R P₁.Ring A)
    (IsScalarTower.toAlgHom R P₂.infinitesimal.Ring A) ⋯ ⋯
this : ∀ r ∈ P₁.ker, lift r ∈ P₂.infinitesimal.ker
r : P₁.Ring
hr : r ∈ P₁.ker
s : P₁.Ring
hs : s ∈ P₁.ker
⊢ lift r * lift s = 0

Tactic:
← Ideal.mem_bot,

Goal state after:
R : Type u
A : Type v
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
B : Type u_1
P : Type u_2
C : Type u_3
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R B
inst✝⁴ : CommRing C
inst✝³ : Algebra R C
inst✝² : CommRing P
inst✝¹ : Algebra R P
P₁ : Extension R A
P₂ : Extension R A
inst✝ : FormallySmooth R P₁.Ring
lift : P₁.Ring →ₐ[R] P₂.infinitesimal.Ring :=
  FormallySmooth.liftOfSurjective (IsScalarTower.toAlgHom R P₁.Ring A)
    (IsScalarTower.toAlgHom R P₂.infinitesimal.Ring A) ⋯ ⋯
this : ∀ r ∈ P₁.ker, lift r ∈ P₂.infinitesimal.ker
r : P₁.Ring
hr : r ∈ P₁.ker
s : P₁.Ring
hs : s ∈ P₁.ker
⊢ lift r * lift s ∈ ⊥
Module: Mathlib.RingTheory.Smooth.Kaehler

Goal state before:
pp: R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g✝ : S →ₐ[R] P
f : P →ₐ[R] S
hf' : RingHom.ker f ^ 2 = ⊥
g : S →ₐ[R] P
hg : f.comp g = AlgHom.id R S
a b : P
⊢ (a - g (f a)) * (b - g (f b)) = 0

Tactic:
← Ideal.mem_bot,

Goal state after:
R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g✝ : S →ₐ[R] P
f : P →ₐ[R] S
hf' : RingHom.ker f ^ 2 = ⊥
g : S →ₐ[R] P
hg : f.comp g = AlgHom.id R S
a b : P
⊢ (a - g (f a)) * (b - g (f b)) ∈ ⊥
Module: Mathlib.RingTheory.Smooth.Kaehler

Goal state before:
pp: case e_a
R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
p : P
s : S
m : ↥(RingHom.ker (algebraMap P S))
⊢ g ((algebraMap P S) p) * ↑m - p * ↑m = 0

Tactic:
← Ideal.mem_bot,

Goal state after:
case e_a
R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
g : S →ₐ[R] P
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hg : (IsScalarTower.toAlgHom R P S).comp g = AlgHom.id R S
this : Algebra S P := g.toAlgebra
p : P
s : S
m : ↥(RingHom.ker (algebraMap P S))
⊢ g ((algebraMap P S) p) * ↑m - p * ↑m ∈ ⊥
Module: Mathlib.RingTheory.Smooth.Kaehler

Goal state before:
pp: R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing P
inst✝² : CommRing S
inst✝¹ : Algebra R P
inst✝ : Algebra P S
l : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S))
hl : l ∘ₗ kerToTensor R P S = LinearMap.id
x y : P
h : (algebraMap P S) x = (algebraMap P S) y
⊢ x - y = ↑(l (1 ⊗ₜ[P] ((D R P) x - (D R P) y)))

Tactic:
← map_sub

Goal state after:
R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing P
inst✝² : CommRing S
inst✝¹ : Algebra R P
inst✝ : Algebra P S
l : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S))
hl : l ∘ₗ kerToTensor R P S = LinearMap.id
x y : P
h : (algebraMap P S) x = (algebraMap P S) y
⊢ x - y = ↑(l (1 ⊗ₜ[P] (D R P) (x - y)))
Module: Mathlib.RingTheory.Smooth.Kaehler

Goal state before:
pp: R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
l : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S))
hl : l ∘ₗ kerToTensor R P S = LinearMap.id
σ : S → P
hσ : ∀ (x : S), (algebraMap P S) (σ x) = x
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
a b : S
x y : S ⊗[P] Ω[P⁄R]
⊢ ↑(l x) * ↑(l y) = 0

Tactic:
← Ideal.mem_bot,

Goal state after:
R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
l : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S))
hl : l ∘ₗ kerToTensor R P S = LinearMap.id
σ : S → P
hσ : ∀ (x : S), (algebraMap P S) (σ x) = x
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
a b : S
x y : S ⊗[P] Ω[P⁄R]
⊢ ↑(l x) * ↑(l y) ∈ ⊥
Module: Mathlib.RingTheory.Smooth.Kaehler

Goal state before:
pp: R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hf : Surjective ⇑(algebraMap P S)
l : { l // l ∘ₗ kerToTensor R P S = LinearMap.id }
p s : P
x y : S ⊗[P] Ω[P⁄R]
⊢ ↑(↑l x) * ↑(↑l y) = 0

Tactic:
← Ideal.mem_bot,

Goal state after:
R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hf : Surjective ⇑(algebraMap P S)
l : { l // l ∘ₗ kerToTensor R P S = LinearMap.id }
p s : P
x y : S ⊗[P] Ω[P⁄R]
⊢ ↑(↑l x) * ↑(↑l y) ∈ ⊥
Module: Mathlib.RingTheory.Smooth.Kaehler

Goal state before:
pp: R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
hf : Surjective ⇑(algebraMap P S)
l : { l // l ∘ₗ kerToTensor R P S = LinearMap.id }
p s : P
this : ∀ (x y : S ⊗[P] Ω[P⁄R]), ↑(↑l x) * ↑(↑l y) = 0
⊢ s * ↑(↑l (1 ⊗ₜ[P] (D R P) p)) - ↑(↑l (1 ⊗ₜ[P] (D R P) s)) * ↑(↑l (1 ⊗ₜ[P] (D R P) p)) =
    ↑(↑l ((algebraMap P S) s ⊗ₜ[P] (D R P) p))

Tactic:
simp only [this, Algebra.algebraMap_eq_smul_one, ← smul_tmul', LinearMapClass.map_smul,
      SetLike.val_smul, smul_eq_mul, sub_zero]

Goal state after: