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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z
a : R
ha : a ∈ I
⊢ l a • (algebraMap R S) a * (algebraMap R S) p ∈ ⇑(algebraMap (↥R[x]) S) '' ↑(Ideal.map (algebraMap R ↥R[x]) I)

Tactic:
smul_eq_mul,

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z
a : R
ha : a ∈ I
⊢ l a * (algebraMap R S) a * (algebraMap R S) p ∈ ⇑(algebraMap (↥R[x]) S) '' ↑(Ideal.map (algebraMap R ↥R[x]) I)
Module: Mathlib.RingTheory.Algebraic.StronglyTranscendental

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✝ : IsReduced S
x : S
hx : IsStronglyTranscendental R x
q : Ideal S
hq : q ∈ minimalPrimes S
u : S
p : R[X]
e : (aeval x) p * u ∈ q
this✝¹ : q.IsPrime
this✝ : Ring.KrullDimLE 0 (Localization.AtPrime q)
this : Field (Localization.AtPrime q) := ⋯.toField
⊢ (algebraMap S (Localization.AtPrime q)) ((aeval x) p * u) = 0

Tactic:
rw [← Ideal.mem_bot, ← IsLocalRing.maximalIdeal_eq_bot,
      ← Localization.AtPrime.map_eq_maximalIdeal]; exact Ideal.mem_map_of_mem _ e

Goal state after:
Module: Mathlib.RingTheory.IntegralClosure.GoingDown

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : IsDomain S
inst✝⁴ : FaithfulSMul R S
inst✝³ : Algebra.IsIntegral R S
inst✝² : IsIntegrallyClosed R
this : IsDomain R
p : Ideal R
inst✝¹ : p.IsPrime
Q : Ideal S
inst✝ : Q.IsPrime
hpQ : p < Ideal.under R Q
SQ : Type u_2 := Localization.AtPrime Q
x : R
hx : x ∈ Ideal.comap (algebraMap R SQ) (Ideal.map (algebraMap R SQ) p)
a : S
ha : x • a ∈ Ideal.map (algebraMap R S) p
haQ : a ∉ Q
f : R[X]
hfm : f.Monic
hfa : (aeval (x • a)) f = 0
hf : ∀ (i : ℕ), i ≠ f.natDegree → f.coeff i ∈ p
g : R[X] := minpoly R a
⊢ ∃ i, i ≠ g.natDegree ∧ g.coeff i ∉ p

Tactic:
by_contra! H

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : IsDomain S
inst✝⁴ : FaithfulSMul R S
inst✝³ : Algebra.IsIntegral R S
inst✝² : IsIntegrallyClosed R
this : IsDomain R
p : Ideal R
inst✝¹ : p.IsPrime
Q : Ideal S
inst✝ : Q.IsPrime
hpQ : p < Ideal.under R Q
SQ : Type u_2 := Localization.AtPrime Q
x : R
hx : x ∈ Ideal.comap (algebraMap R SQ) (Ideal.map (algebraMap R SQ) p)
a : S
ha : x • a ∈ Ideal.map (algebraMap R S) p
haQ : a ∉ Q
f : R[X]
hfm : f.Monic
hfa : (aeval (x • a)) f = 0
hf : ∀ (i : ℕ), i ≠ f.natDegree → f.coeff i ∈ p
g : R[X] := minpoly R a
H : ∀ (i : ℕ), i ≠ g.natDegree → g.coeff i ∈ p
⊢ False
Module: Mathlib.RingTheory.Conductor

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
h_alg : Function.Injective ⇑(algebraMap (↥R[x]) S)
z : S
hz : z ∈ R[x]
hy : ⟨z, hz⟩ ∈ comap (algebraMap (↥R[x]) S) (Ideal.map (algebraMap R S) I)
p : R
hp : p ∈ comap (algebraMap R S) (conductor R x)
q : R
hq : q ∈ I
hpq : p + q = 1
temp : (algebraMap R S) p * z + (algebraMap R S) q * z = z
this : z ∈ ⇑(algebraMap (↥R[x]) S) '' ↑(Ideal.map (algebraMap R ↥R[x]) I) ↔ ⟨z, hz⟩ ∈ Ideal.map (algebraMap R ↥R[x]) I
⊢ ⟨z, hz⟩ ∈ Ideal.map (algebraMap R ↥R[x]) I

Tactic:
← this,

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
h_alg : Function.Injective ⇑(algebraMap (↥R[x]) S)
z : S
hz : z ∈ R[x]
hy : ⟨z, hz⟩ ∈ comap (algebraMap (↥R[x]) S) (Ideal.map (algebraMap R S) I)
p : R
hp : p ∈ comap (algebraMap R S) (conductor R x)
q : R
hq : q ∈ I
hpq : p + q = 1
temp : (algebraMap R S) p * z + (algebraMap R S) q * z = z
this : z ∈ ⇑(algebraMap (↥R[x]) S) '' ↑(Ideal.map (algebraMap R ↥R[x]) I) ↔ ⟨z, hz⟩ ∈ Ideal.map (algebraMap R ↥R[x]) I
⊢ z ∈ ⇑(algebraMap (↥R[x]) S) '' ↑(Ideal.map (algebraMap R ↥R[x]) I)
Module: Mathlib.RingTheory.Conductor

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
h_alg : Function.Injective ⇑(algebraMap (↥R[x]) S)
z : S
hz : z ∈ R[x]
hy : ⟨z, hz⟩ ∈ comap (algebraMap (↥R[x]) S) (Ideal.map (algebraMap R S) I)
p : R
hp : p ∈ comap (algebraMap R S) (conductor R x)
q : R
hq : q ∈ I
hpq : p + q = 1
temp : (algebraMap R S) p * z + (algebraMap R S) q * z = z
h : z ∈ ⇑(algebraMap (↥R[x]) S) '' ↑(Ideal.map (algebraMap R ↥R[x]) I)
x₁ : ↥R[x]
hx₁ : x₁ ∈ ↑(Ideal.map (algebraMap R ↥R[x]) I)
hx₂ : (algebraMap (↥R[x]) S) x₁ = z
⊢ z = z • 1

Tactic:
smul_eq_mul,

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
h_alg : Function.Injective ⇑(algebraMap (↥R[x]) S)
z : S
hz : z ∈ R[x]
hy : ⟨z, hz⟩ ∈ comap (algebraMap (↥R[x]) S) (Ideal.map (algebraMap R S) I)
p : R
hp : p ∈ comap (algebraMap R S) (conductor R x)
q : R
hq : q ∈ I
hpq : p + q = 1
temp : (algebraMap R S) p * z + (algebraMap R S) q * z = z
h : z ∈ ⇑(algebraMap (↥R[x]) S) '' ↑(Ideal.map (algebraMap R ↥R[x]) I)
x₁ : ↥R[x]
hx₁ : x₁ ∈ ↑(Ideal.map (algebraMap R ↥R[x]) I)
hx₂ : (algebraMap (↥R[x]) S) x₁ = z
⊢ z = z * 1
Module: Mathlib.FieldTheory.RatFunc.AsPolynomial

Goal state before:
pp: K : Type u
inst✝¹ : Field K
L : Type u
inst✝ : Field L
f : K →+* L
a : L
c : K
⊢ eval f a (C c) = f c

Tactic:
by simp [eval]

Goal state after:
Module: Mathlib.FieldTheory.RatFunc.AsPolynomial

Goal state before:
pp: K : Type u
inst✝¹ : Field K
L : Type u
inst✝ : Field L
f : K →+* L
a : L
⊢ eval f a X = a

Tactic:
by simp [eval]

Goal state after:
Module: Mathlib.FieldTheory.RatFunc.AsPolynomial

Goal state before:
pp: K : Type u
inst✝¹ : Field K
L : Type u
inst✝ : Field L
f : K →+* L
a : L
⊢ eval f a 0 = 0

Tactic:
by simp [eval]

Goal state after:
Module: Mathlib.FieldTheory.RatFunc.AsPolynomial

Goal state before:
pp: K : Type u
inst✝¹ : Field K
L : Type u
inst✝ : Field L
f : K →+* L
a : L
⊢ eval f a 1 = 1

Tactic:
by simp [eval]

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R[X]
q : S[X]
hp : p.Monic
hq : q.Monic
H : q ∣ map (algebraMap R S) p
i : ℕ
a✝ : Nontrivial S
T : Type u_2
w✝⁴ : CommRing T
w✝³ : Algebra S T
w✝² : Module.Finite S T
w✝¹ : Module.Free S T
w✝ : Nontrivial T
hqT : (map (algebraMap S T) q).Splits
algInst✝ : Algebra R T := ((algebraMap S T).comp (algebraMap R S)).toAlgebra
scalarTowerInst✝ : IsScalarTower R S T := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap R T))
⊢ IsIntegral R ((IsScalarTower.toAlgHom R S T) (q.coeff i))

Tactic:
IsScalarTower.coe_toAlgHom',

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R[X]
q : S[X]
hp : p.Monic
hq : q.Monic
H : q ∣ map (algebraMap R S) p
i : ℕ
a✝ : Nontrivial S
T : Type u_2
w✝⁴ : CommRing T
w✝³ : Algebra S T
w✝² : Module.Finite S T
w✝¹ : Module.Free S T
w✝ : Nontrivial T
hqT : (map (algebraMap S T) q).Splits
algInst✝ : Algebra R T := ((algebraMap S T).comp (algebraMap R S)).toAlgebra
scalarTowerInst✝ : IsScalarTower R S T := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap R T))
⊢ IsIntegral R ((algebraMap S T) (q.coeff i))
Module: Mathlib.RingTheory.Polynomial.IsIntegral

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 : IsDomain R
inst✝ : FaithfulSMul R S
i : ℕ
H : ∀ {q : S[X]}, IsAlmostIntegral R[X] q → IsAlmostIntegral R q.leadingCoeff
n : ℕ
IH : ∀ m < n, ∀ {p : S[X]}, IsAlmostIntegral R[X] p → p.natDegree = m → IsAlmostIntegral R (p.coeff i)
p : S[X]
hp : IsAlmostIntegral R[X] p
hn : p.natDegree = n
hp' : ¬p.natDegree = 0
hi : ¬i = p.natDegree
⊢ IsAlmostIntegral R[X] p.eraseLead

Tactic:
rw [← self_sub_monomial_natDegree_leadingCoeff, ← mem_completeIntegralClosure,
      ← C_mul_X_pow_eq_monomial, ← map_X (algebraMap R S), ← Polynomial.map_pow]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsDomain R
inst✝ : FaithfulSMul R S
i : ℕ
H : ∀ {q : S[X]}, IsAlmostIntegral R[X] q → IsAlmostIntegral R q.leadingCoeff
n : ℕ
IH : ∀ m < n, ∀ {p : S[X]}, IsAlmostIntegral R[X] p → p.natDegree = m → IsAlmostIntegral R (p.coeff i)
p : S[X]
hp : IsAlmostIntegral R[X] p
hn : p.natDegree = n
hp' : ¬p.natDegree = 0
hi : ¬i = p.natDegree
⊢ p - C p.leadingCoeff * map (algebraMap R S) (X ^ p.natDegree) ∈ completeIntegralClosure R[X] S[X]
Module: Mathlib.RingTheory.Polynomial.IsIntegral

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : S[X]
hp : IsIntegral R[X] p
i✝ : ℕ
a✝¹ : Nontrivial R
a✝ : Nontrivial S
hp0 : p ≠ 0
q : R[X][X] := minpoly R[X] p
m : ℕ := (q.support.sup fun i ↦ (q.coeff i).natDegree) + p.natDegree + 1
hm₁ : ∀ (i : ℕ), (q.coeff i).natDegree < m
i : ℕ
hi : i < q.natDegree
⊢ (q.coeff i).natDegree + (X ^ (m * i)).natDegree < m * q.natDegree

Tactic:
natDegree_X_pow,

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : S[X]
hp : IsIntegral R[X] p
i✝ : ℕ
a✝¹ : Nontrivial R
a✝ : Nontrivial S
hp0 : p ≠ 0
q : R[X][X] := minpoly R[X] p
m : ℕ := (q.support.sup fun i ↦ (q.coeff i).natDegree) + p.natDegree + 1
hm₁ : ∀ (i : ℕ), (q.coeff i).natDegree < m
i : ℕ
hi : i < q.natDegree
⊢ (q.coeff i).natDegree + m * i < m * q.natDegree
Module: Mathlib.RingTheory.Polynomial.IsIntegral

Goal state before:
pp: case e'_6
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
σ : Type w
hσ : Finite σ
f : MvPolynomial PEmpty.{w + 1} S
H : (algebraMap (MvPolynomial PEmpty.{w + 1} R) (MvPolynomial PEmpty.{w + 1} S)).IsIntegralElem f
n : PEmpty.{w + 1} →₀ ℕ
⊢ (isEmptyAlgEquiv S PEmpty.{w + 1}).symm.toRingEquiv.toRingHom (coeff n f) = f

Tactic:
obtain rfl := Subsingleton.elim n 0
      have : constantCoeff = (isEmptyAlgEquiv S PEmpty).toRingHom := by aesop
      simpa [-EmbeddingLike.apply_eq_iff_eq, -isEmptyAlgEquiv_apply] using!
        congr((isEmptyAlgEquiv S PEmpty.{w + 1}).symm ($this f))

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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
P : Ideal R
inst✝ : P.IsPrime
I : Ideal R[X]
p : P.ResidueField[X]
hp : Ideal.map (mapRingHom (algebraMap R P.ResidueField)) I = Ideal.span {p}
this✝¹ : Algebra (R ⧸ P)[X] P.ResidueField[X] := (mapRingHom (algebraMap (R ⧸ P) P.ResidueField)).toAlgebra
this✝ : IsLocalization (Submonoid.map C (R ⧸ P)⁰) P.ResidueField[X]
this : p ∈ Ideal.map (algebraMap (R ⧸ P)[X] P.ResidueField[X]) (Ideal.map (mapRingHom (algebraMap R (R ⧸ P))) I)
⊢ ∃ p ∈ I, Ideal.span {map (algebraMap R P.ResidueField) p} = Ideal.map (mapRingHom (algebraMap R P.ResidueField)) I

Tactic:
obtain ⟨⟨⟨r, hr⟩, s⟩, e⟩ := (IsLocalization.mem_map_algebraMap_iff ((R ⧸ P)⁰.map C) _).mp this

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
P : Ideal R
inst✝ : P.IsPrime
I : Ideal R[X]
p : P.ResidueField[X]
hp : Ideal.map (mapRingHom (algebraMap R P.ResidueField)) I = Ideal.span {p}
this✝¹ : Algebra (R ⧸ P)[X] P.ResidueField[X] := (mapRingHom (algebraMap (R ⧸ P) P.ResidueField)).toAlgebra
this✝ : IsLocalization (Submonoid.map C (R ⧸ P)⁰) P.ResidueField[X]
this : p ∈ Ideal.map (algebraMap (R ⧸ P)[X] P.ResidueField[X]) (Ideal.map (mapRingHom (algebraMap R (R ⧸ P))) I)
s : ↥(Submonoid.map C (R ⧸ P)⁰)
r : (R ⧸ P)[X]
hr : r ∈ Ideal.map (mapRingHom (algebraMap R (R ⧸ P))) I
e :
  p * (algebraMap (R ⧸ P)[X] P.ResidueField[X]) ↑(⟨r, hr⟩, s).2 =
    (algebraMap (R ⧸ P)[X] P.ResidueField[X]) ↑(⟨r, hr⟩, s).1
⊢ ∃ p ∈ I, Ideal.span {map (algebraMap R P.ResidueField) p} = Ideal.map (mapRingHom (algebraMap R P.ResidueField)) I
Module: Mathlib.FieldTheory.RatFunc.Basic

Goal state before:
pp: case H
K : Type u
inst✝ : CommRing K
r : K
y✝ : K[X] × ↥K[X]⁰
⊢ { toFractionRing := Localization.mk (r • y✝.1) y✝.2 } =
    { toFractionRing := Localization.mk (Polynomial.C r • y✝.1) y✝.2 }

Tactic:
smul_eq_mul,

Goal state after:
case H
K : Type u
inst✝ : CommRing K
r : K
y✝ : K[X] × ↥K[X]⁰
⊢ { toFractionRing := Localization.mk (r • y✝.1) y✝.2 } =
    { toFractionRing := Localization.mk (Polynomial.C r * y✝.1) y✝.2 }
Module: Mathlib.FieldTheory.RatFunc.Basic

Goal state before:
pp: K : Type u
inst✝⁵ : CommRing K
inst✝⁴ : IsDomain K
R : Type u_1
F : Type u_2
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : FunLike F K[X] R[X]
inst✝ : MonoidHomClass F K[X] R[X]
φ : F
hφ : K[X]⁰ ≤ Submonoid.comap φ R[X]⁰
p q : K[X]
hq : q ≠ 0
hq' : φ q ≠ 0
⊢ (map φ hφ) ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q) =
    (algebraMap R[X] R⟮X⟯) (φ p) / (algebraMap R[X] R⟮X⟯) (φ q)

Tactic:
simp only [← mk_eq_div, mk_eq_localization_mk _ hq, map_apply_ofFractionRing_mk,
    mk_eq_localization_mk _ hq']

Goal state after:
Module: Mathlib.RingTheory.Etale.StandardEtale

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : StandardEtalePair R
I : Ideal S
hI : I ^ 2 = ⊥
x : S
hx : P.HasMap ((Ideal.Quotient.mk I) x)
⊢ ∃! ε, ε ∈ I ∧ P.HasMap (x + ε)

Tactic:
by
  have hf := Ideal.Quotient.eq_zero_iff_mem.mp
    ((aeval_algHom_apply (Ideal.Quotient.mkₐ R I) _ _).symm.trans hx.1)
  obtain ⟨⟨_, a, ha, -⟩, rfl⟩ := hx.2
  obtain ⟨a, rfl⟩ := Ideal.Quotient.mk_surjective a
  simp_rw [← Ideal.Quotient.mkₐ_eq_mk R, aeval_algHom_apply, ← map_mul, ← map_one
    (Ideal.Quotient.mkₐ R I), Ideal.Quotient.mkₐ_eq_mk, Ideal.Quotient.mk_eq_mk_iff_sub_mem] at ha
  obtain ⟨p₁, p₂, n, e⟩ := P.cond
  apply_fun aeval x at e
  simp only [map_add, map_mul, map_pow] at e
  obtain ⟨ε, hεI, b, hb⟩ : ∃ ε ∈ I, ∃ b, aeval x (derivative P.f) * b = 1 + ε := by
    refine ⟨_, ?_, (a ^ n * aeval x p₁), sub_eq_iff_eq_add'.mp rfl⟩
    convert_to (aeval x P.g * a) ^ n - 1 - aeval x P.f * (a ^ n * aeval x p₂) ∈ I
    · linear_combination a ^ n * e
    · exact sub_mem (Ideal.mem_of_dvd _ (sub_one_dvd_pow_sub_one _ _) ha) (I.mul_mem_right _ hf)
  have : aeval x P.f ^ 2 = 0 := hI.le (Ideal.pow_mem_pow hf 2)
  have : aeval x P.f * ε = 0 := ((pow_two _).symm.trans hI).le (Ideal.mul_mem_mul hf hεI)
  refine ⟨aeval x P.f * -b, ⟨I.mul_mem_right _ hf, ?_, ?_⟩, ?_⟩
  · rw [Polynomial.aeval_add_of_sq_eq_zero _ _ _ (by grind)]; grind
  · rw [← IsNilpotent.isUnit_quotient_mk_iff (I := I) ⟨2, hI⟩, ← Ideal.Quotient.mkₐ_eq_mk R,
      ← aeval_algHom_apply, Ideal.Quotient.mkₐ_eq_mk, map_add,
      Ideal.Quotient.eq_zero_iff_mem.mpr (I.mul_mem_right _ hf), add_zero]
    exact hx.2
  · rintro ε' ⟨hε'I, hε', hε''⟩
    rw [Polynomial.aeval_add_of_sq_eq_zero _ _ _ (hI.le (Ideal.pow_mem_pow hε'I 2))] at hε'
    have : ε * ε' = 0 := ((pow_two _).symm.trans hI).le (Ideal.mul_mem_mul hεI hε'I)
    grind

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

Goal state before:
pp: K : Type u
inst✝ : Field K
x y : K⟮X⟯
hxy : x + y ≠ 0
h_zero : x.num * y.denom + x.denom * y.num = 0
h : (x + y).num * (x.denom * y.denom) = (x.num * y.denom + x.denom * y.num) * (x + y).denom
⊢ False

Tactic:
rw [h_zero, zero_mul] at h

Goal state after:
K : Type u
inst✝ : Field K
x y : K⟮X⟯
hxy : x + y ≠ 0
h_zero : x.num * y.denom + x.denom * y.num = 0
h : (x + y).num * (x.denom * y.denom) = 0
⊢ False
Module: Mathlib.RingTheory.Etale.StandardEtale

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
P : StandardEtalePair R
⊢ IsUnit
    ((aeval ((algebraMap (AdjoinRoot P.f) (Localization.Away ((AdjoinRoot.mk P.f) P.g))) (AdjoinRoot.root P.f))) P.g)

pp: case refine_3
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
P : StandardEtalePair R
⊢ (P.lift ((algebraMap (AdjoinRoot P.f) (Localization.Away ((AdjoinRoot.mk P.f) P.g))) (AdjoinRoot.root P.f)) ⋯).comp
      (IsLocalization.Away.liftAlgHom ((AdjoinRoot.mk P.f) P.g) ⋯) =
    AlgHom.id R (Localization.Away ((AdjoinRoot.mk P.f) P.g))

pp: case refine_4
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
P : StandardEtalePair R
⊢ (IsLocalization.Away.liftAlgHom ((AdjoinRoot.mk P.f) P.g) ⋯).comp
      (P.lift ((algebraMap (AdjoinRoot P.f) (Localization.Away ((AdjoinRoot.mk P.f) P.g))) (AdjoinRoot.root P.f)) ⋯) =
    AlgHom.id R P.Ring

Tactic:
· rw [aeval_algebraMap_apply, AdjoinRoot.aeval_eq]
    exact IsLocalization.Away.algebraMap_isUnit ..

Goal state after:
case refine_3
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
P : StandardEtalePair R
⊢ (P.lift ((algebraMap (AdjoinRoot P.f) (Localization.Away ((AdjoinRoot.mk P.f) P.g))) (AdjoinRoot.root P.f)) ⋯).comp
      (IsLocalization.Away.liftAlgHom ((AdjoinRoot.mk P.f) P.g) ⋯) =
    AlgHom.id R (Localization.Away ((AdjoinRoot.mk P.f) P.g))

case refine_4
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
P : StandardEtalePair R
⊢ (IsLocalization.Away.liftAlgHom ((AdjoinRoot.mk P.f) P.g) ⋯).comp
      (P.lift ((algebraMap (AdjoinRoot P.f) (Localization.Away ((AdjoinRoot.mk P.f) P.g))) (AdjoinRoot.root P.f)) ⋯) =
    AlgHom.id R P.Ring
Module: Mathlib.RingTheory.Smooth.IntegralClosure

Goal state before:
pp: R : Type u_1
S : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing B
inst✝ : Algebra R B
φ : S[X] →ₐ[R] B
hφ : Function.Surjective ⇑φ
f : S[X]
hf : f.Monic
hf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)
hfx : RingHom.ker φ.toRingHom = Ideal.span {f}
h✝ : Nontrivial B
hfd : f.natDegree ≠ 0
algInst✝¹ : Algebra S B := (φ.comp C).toAlgebra
this : Nontrivial S
y : S[X]
hy : IsIntegral R (φ y)
S' : Type u_2
w✝⁴ : CommRing S'
w✝³ : Algebra S S'
w✝² : Module.Finite S S'
w✝¹ : Module.Free S S'
w✝ : Nontrivial S'
hS' : (Polynomial.map (algebraMap S S') f).Splits
m : Multiset S'
hm : Polynomial.map (algebraMap S S') f = (Multiset.map (fun x ↦ X - C x) m).prod
algInst✝ : Algebra R S' := ((algebraMap S S').comp (algebraMap R S)).toAlgebra
scalarTowerInst✝ : IsScalarTower R S S' := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap R S'))
hm' : ∀ a ∈ m, IsIntegral R a
hmc : m.card = f.natDegree
a : S'
ham : a ∈ m
⊢ (Multiset.map (fun x ↦ X - C x) (a ::ₘ m.erase a)).prod ∣
    (Multiset.map (fun x ↦ X - C x) (m.erase a)).prod * (Polynomial.map (algebraMap S S') y - C ((aeval a) y))

Tactic:
rw [Multiset.map_cons, Multiset.prod_cons, mul_comm]

Goal state after:
R : Type u_1
S : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing B
inst✝ : Algebra R B
φ : S[X] →ₐ[R] B
hφ : Function.Surjective ⇑φ
f : S[X]
hf : f.Monic
hf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)
hfx : RingHom.ker φ.toRingHom = Ideal.span {f}
h✝ : Nontrivial B
hfd : f.natDegree ≠ 0
algInst✝¹ : Algebra S B := (φ.comp C).toAlgebra
this : Nontrivial S
y : S[X]
hy : IsIntegral R (φ y)
S' : Type u_2
w✝⁴ : CommRing S'
w✝³ : Algebra S S'
w✝² : Module.Finite S S'
w✝¹ : Module.Free S S'
w✝ : Nontrivial S'
hS' : (Polynomial.map (algebraMap S S') f).Splits
m : Multiset S'
hm : Polynomial.map (algebraMap S S') f = (Multiset.map (fun x ↦ X - C x) m).prod
algInst✝ : Algebra R S' := ((algebraMap S S').comp (algebraMap R S)).toAlgebra
scalarTowerInst✝ : IsScalarTower R S S' := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap R S'))
hm' : ∀ a ∈ m, IsIntegral R a
hmc : m.card = f.natDegree
a : S'
ham : a ∈ m
⊢ (Multiset.map (fun x ↦ X - C x) (m.erase a)).prod * (X - C a) ∣
    (Multiset.map (fun x ↦ X - C x) (m.erase a)).prod * (Polynomial.map (algebraMap S S') y - C ((aeval a) y))
Module: Mathlib.RingTheory.Unramified.LocalStructure

Goal state before:
pp: case refine_1
R : Type u_1
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
Q : Ideal S
inst✝² : Q.IsPrime
inst✝¹ : Module.Finite R S
inst✝ : IsUnramifiedAt R Q
this✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q) :=
  Localization.AtPrime.algebraOfLiesOver (Ideal.under R Q) Q
t : S
htQ : t ∉ Q
htQ' : ∀ Q' ∈ (Ideal.under R Q).primesOver S, Q' ≠ Q → t ∈ Q'
ht : (Ideal.under R Q).ResidueField[(algebraMap S Q.ResidueField) t] = ⊤
p : Ideal R := Ideal.under R Q
this✝ : Algebra (Localization.AtPrime p) (Localization.AtPrime (Ideal.under (↥R[t]) Q)) :=
  Localization.AtPrime.algebraOfLiesOver p (Ideal.under (↥R[t]) Q)
this : Algebra (Localization.AtPrime (Ideal.under (↥R[t]) Q)) (Localization.AtPrime Q) :=
  Localization.AtPrime.algebraOfLiesOver (Ideal.under (↥R[t]) Q) Q
Q' : Ideal S
x✝ : Q' ∈ (Ideal.under (↥R[t]) Q).primesOver S
left✝ : Q'.IsPrime
right✝ : Q'.LiesOver (Ideal.under (↥R[t]) Q)
⊢ Q' ∈ {Q}

Tactic:
by_contra! H

Goal state after:
case refine_1
R : Type u_1
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
Q : Ideal S
inst✝² : Q.IsPrime
inst✝¹ : Module.Finite R S
inst✝ : IsUnramifiedAt R Q
this✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q) :=
  Localization.AtPrime.algebraOfLiesOver (Ideal.under R Q) Q
t : S
htQ : t ∉ Q
htQ' : ∀ Q' ∈ (Ideal.under R Q).primesOver S, Q' ≠ Q → t ∈ Q'
ht : (Ideal.under R Q).ResidueField[(algebraMap S Q.ResidueField) t] = ⊤
p : Ideal R := Ideal.under R Q
this✝ : Algebra (Localization.AtPrime p) (Localization.AtPrime (Ideal.under (↥R[t]) Q)) :=
  Localization.AtPrime.algebraOfLiesOver p (Ideal.under (↥R[t]) Q)
this : Algebra (Localization.AtPrime (Ideal.under (↥R[t]) Q)) (Localization.AtPrime Q) :=
  Localization.AtPrime.algebraOfLiesOver (Ideal.under (↥R[t]) Q) Q
Q' : Ideal S
x✝ : Q' ∈ (Ideal.under (↥R[t]) Q).primesOver S
left✝ : Q'.IsPrime
right✝ : Q'.LiesOver (Ideal.under (↥R[t]) Q)
H : Q' ∉ {Q}
⊢ False
Module: Mathlib.RingTheory.Unramified.LocalStructure

Goal state before:
pp: R : Type u_1
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
Q : Ideal S
inst✝² : Q.IsPrime
inst✝¹ : Module.Finite R S
inst✝ : IsUnramifiedAt R Q
x : S
hQ' : (Ideal.under (↥R[x]) Q).primesOver S = {Q}
hQ'Q : Function.Bijective ⇑(algebraMap (Ideal.under (↥R[x]) Q).ResidueField Q.ResidueField)
S' : Subalgebra R S := R[x]
Q' : Ideal ↥S' := Ideal.under (↥S') Q
this : Algebra (Localization.AtPrime Q') (Localization.AtPrime Q) := Localization.AtPrime.algebraOfLiesOver Q' Q
⊢ ∃ f ∉ Q, HasStandardEtaleSurjectionOn R f

Tactic:
have : Module.Finite S' S := .of_restrictScalars_finite R _ _

Goal state after:
R : Type u_1
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
Q : Ideal S
inst✝² : Q.IsPrime
inst✝¹ : Module.Finite R S
inst✝ : IsUnramifiedAt R Q
x : S
hQ' : (Ideal.under (↥R[x]) Q).primesOver S = {Q}
hQ'Q : Function.Bijective ⇑(algebraMap (Ideal.under (↥R[x]) Q).ResidueField Q.ResidueField)
S' : Subalgebra R S := R[x]
Q' : Ideal ↥S' := Ideal.under (↥S') Q
this✝ : Algebra (Localization.AtPrime Q') (Localization.AtPrime Q) := Localization.AtPrime.algebraOfLiesOver Q' Q
this : Module.Finite (↥S') S
⊢ ∃ f ∉ Q, HasStandardEtaleSurjectionOn R f