mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.RamificationInertia.Ramification

Goal state before:
pp: case neg
R : Type u
inst✝³ : CommRing R
S : Type v
inst✝² : CommRing S
inst✝¹ : Algebra R S
p : Ideal R
P : Ideal S
inst✝ : IsDedekindDomain S
hp : map f p ≠ ⊥
hP : P.IsPrime
hP₂ : ¬P = ⊥
⊢ emultiplicity P (map f p) = emultiplicity P (map f p)

pp: R : Type u
inst✝³ : CommRing R
S : Type v
inst✝² : CommRing S
inst✝¹ : Algebra R S
p : Ideal R
P : Ideal S
inst✝ : IsDedekindDomain S
hp : map f p ≠ ⊥
hP : P.IsPrime
hP₂ : ¬P = ⊥
⊢ Irreducible P

Tactic:
]

Goal state after:
R : Type u
inst✝³ : CommRing R
S : Type v
inst✝² : CommRing S
inst✝¹ : Algebra R S
p : Ideal R
P : Ideal S
inst✝ : IsDedekindDomain S
hp : map f p ≠ ⊥
hP : P.IsPrime
hP₂ : ¬P = ⊥
⊢ Irreducible P
Module: Mathlib.NumberTheory.RamificationInertia.Ramification

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
S : Type v
inst✝² : CommRing S
inst✝¹ : Algebra R S
p : Ideal R
P : Ideal S
inst✝ : IsDedekindDomain S
hp0 : map f p ≠ ⊥
hP : P.IsPrime
hP0 : P ≠ ⊥
⊢ Multiset.count P (normalizedFactors (map f p)) = Multiset.count P (normalizedFactors (map f p))

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Ramification

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
S : Type v
inst✝² : CommRing S
inst✝¹ : Algebra R S
p : Ideal R
P : Ideal S
inst✝ : IsDedekindDomain S
hp0 : map f p ≠ ⊥
hP : P.IsPrime
le : map f p ≤ P
hP0 : P ≠ ⊥
hPirr : Irreducible P
⊢ Multiset.count P (normalizedFactors (map f p)) ≠ 0

Tactic:
]

Goal state after:
R : Type u
inst✝³ : CommRing R
S : Type v
inst✝² : CommRing S
inst✝¹ : Algebra R S
p : Ideal R
P : Ideal S
inst✝ : IsDedekindDomain S
hp0 : map f p ≠ ⊥
hP : P.IsPrime
le : map f p ≤ P
hP0 : P ≠ ⊥
hPirr : Irreducible P
⊢ Multiset.count P (normalizedFactors (map f p)) ≠ 0
Module: Mathlib.NumberTheory.RamificationInertia.Ramification

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
S : Type v
inst✝² : CommRing S
inst✝¹ : Algebra R S
p : Ideal R
P : Ideal S
inst✝ : IsDedekindDomain S
hp0 : map f p ≠ ⊥
hP : P.IsPrime
le : map f p ≤ P
hP0 : P ≠ ⊥
hPirr : Irreducible P
P' : Ideal S
hP' : P' ∈ normalizedFactors (map f p)
P'_eq : Associated P P'
⊢ P' ∈ normalizedFactors (map f p)

Tactic:
]

Goal state after:
R : Type u
inst✝³ : CommRing R
S : Type v
inst✝² : CommRing S
inst✝¹ : Algebra R S
p : Ideal R
P : Ideal S
inst✝ : IsDedekindDomain S
hp0 : map f p ≠ ⊥
hP : P.IsPrime
le : map f p ≤ P
hP0 : P ≠ ⊥
hPirr : Irreducible P
P' : Ideal S
hP' : P' ∈ normalizedFactors (map f p)
P'_eq : Associated P P'
⊢ P' ∈ normalizedFactors (map f p)
Module: Mathlib.NumberTheory.RamificationInertia.Ramification

Goal state before:
pp: R : Type u
inst✝⁴ : CommRing R
S : Type v
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsDedekindDomain S
p : Ideal R
P : Ideal S
inst✝ : P.IsPrime
hp : P ≠ ⊥
hpP : map f p ≤ P
Sₚ : Type v := Localization.AtPrime P
⊢ ¬map f p ≤ P * P → map (algebraMap R (Localization.AtPrime P)) p = maximalIdeal (Localization.AtPrime P)

Tactic:
]

Goal state after:
R : Type u
inst✝⁴ : CommRing R
S : Type v
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsDedekindDomain S
p : Ideal R
P : Ideal S
inst✝ : P.IsPrime
hp : P ≠ ⊥
hpP : map f p ≤ P
Sₚ : Type v := Localization.AtPrime P
⊢ ¬map f p ≤ P * P → map (algebraMap R (Localization.AtPrime P)) p = maximalIdeal (Localization.AtPrime P)
Module: Mathlib.NumberTheory.RamificationInertia.Ramification

Goal state before:
pp: R : Type u
inst✝⁴ : CommRing R
S : Type v
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsDedekindDomain S
p : Ideal R
P : Ideal S
inst✝ : P.IsPrime
hp : P ≠ ⊥
hpP : map f p ≤ P
Sₚ : Type v := Localization.AtPrime P
H₁ : ¬map f p ≤ P * P
a : Ideal S
ha : map f p = P * a
ha' : ¬a ≤ P
⊢ maximalIdeal (Localization P.primeCompl) * map (algebraMap S (Localization.AtPrime P)) a =
    maximalIdeal (Localization.AtPrime P)

Tactic:
]

Goal state after:
R : Type u
inst✝⁴ : CommRing R
S : Type v
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsDedekindDomain S
p : Ideal R
P : Ideal S
inst✝ : P.IsPrime
hp : P ≠ ⊥
hpP : map f p ≤ P
Sₚ : Type v := Localization.AtPrime P
H₁ : ¬map f p ≤ P * P
a : Ideal S
ha : map f p = P * a
ha' : ¬a ≤ P
⊢ maximalIdeal (Localization P.primeCompl) * map (algebraMap S (Localization.AtPrime P)) a =
    maximalIdeal (Localization.AtPrime P)
Module: Mathlib.NumberTheory.RamificationInertia.Ramification

Goal state before:
pp: case e'_2.e'_6
R : Type u
inst✝⁴ : CommRing R
S : Type v
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsDedekindDomain S
p : Ideal R
P : Ideal S
inst✝ : P.IsPrime
hp : P ≠ ⊥
hpP : map f p ≤ P
Sₚ : Type v := Localization.AtPrime P
H₁ : ¬map f p ≤ P * P
a : Ideal S
ha : map f p = P * a
ha' : ¬a ≤ P
⊢ ¬Disjoint ↑P.primeCompl ↑a

Tactic:
]

Goal state after:
case e'_2.e'_6
R : Type u
inst✝⁴ : CommRing R
S : Type v
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsDedekindDomain S
p : Ideal R
P : Ideal S
inst✝ : P.IsPrime
hp : P ≠ ⊥
hpP : map f p ≤ P
Sₚ : Type v := Localization.AtPrime P
H₁ : ¬map f p ≤ P * P
a : Ideal S
ha : map f p = P * a
ha' : ¬a ≤ P
⊢ ¬Disjoint ↑P.primeCompl ↑a
Module: Mathlib.NumberTheory.RamificationInertia.Ramification

Goal state before:
pp: R : Type u
inst✝⁵ : CommRing R
S : Type v
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : IsDedekindDomain S
inst✝¹ : IsDedekindDomain R
v : Ideal R
w : Ideal S
p : Ideal R
hv : Irreducible v
hp : Prime p
hvp : v ≠ p
inst✝ : w.LiesOver v
h : p ≤ v
⊢ p = v

Tactic:
]

Goal state after:
R : Type u
inst✝⁵ : CommRing R
S : Type v
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : IsDedekindDomain S
inst✝¹ : IsDedekindDomain R
v : Ideal R
w : Ideal S
p : Ideal R
hv : Irreducible v
hp : Prime p
hvp : v ≠ p
inst✝ : w.LiesOver v
h : p ≤ v
⊢ p = v
Module: Mathlib.NumberTheory.RamificationInertia.Ramification

Goal state before:
pp: case neg
R : Type u
inst✝⁶ : CommRing R
S : Type v
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsDedekindDomain S
inst✝² : IsDedekindDomain R
inst✝¹ : FaithfulSMul R S
v : Ideal R
w : Ideal S
p : Ideal R
hv : Irreducible v
hp : Prime p
hw : Irreducible w
hw_bot : w ≠ ⊥
inst✝ : w.LiesOver v
hp_bot : map f p ≠ ⊥
hvp : ¬v = p
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Ramification

Goal state before:
pp: case h₃
R : Type u
inst✝⁶ : CommRing R
S : Type v
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsDedekindDomain S
inst✝² : IsDedekindDomain R
inst✝¹ : FaithfulSMul R S
v : Ideal R
w : Ideal S
hv : Irreducible v
hw : Irreducible w
hw_bot : w ≠ ⊥
inst✝ : w.LiesOver v
I p : Ideal R
hI : I ≠ 0
hp : Prime p
IH : I ≠ ⊥ → emultiplicity w (map f I) = ↑(v.ramificationIdx' w) * emultiplicity v I
h : p * I ≠ ⊥
⊢ ↑(v.ramificationIdx' w) * emultiplicity v p + ↑(v.ramificationIdx' w) * emultiplicity v I =
    ↑(v.ramificationIdx' w) * emultiplicity v p + ↑(v.ramificationIdx' w) * emultiplicity v I

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Ramification

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : CommRing T
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
inst✝¹ : IsDedekindDomain S
inst✝ : IsDedekindDomain T
p : Ideal R
P : Ideal S
Q : Ideal T
hpm : P.IsPrime
hqm : Q.IsPrime
hg0 : map (algebraMap S T) P ≠ ⊥
hfg : map (algebraMap S T) (map f p) ≠ ⊥
hg : map (algebraMap S T) P ≤ Q
⊢ map f p ≠ ⊥

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : CommRing T
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
inst✝¹ : IsDedekindDomain S
inst✝ : IsDedekindDomain T
p : Ideal R
P : Ideal S
Q : Ideal T
hpm : P.IsPrime
hqm : Q.IsPrime
hg0 : map (algebraMap S T) P ≠ ⊥
hfg : map (algebraMap S T) (map f p) ≠ ⊥
hg : map (algebraMap S T) P ≤ Q
⊢ map f p ≠ ⊥
Module: Mathlib.NumberTheory.RamificationInertia.Ramification

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : CommRing T
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
inst✝¹ : IsDedekindDomain S
inst✝ : IsDedekindDomain T
p : Ideal R
P : Ideal S
Q : Ideal T
hpm : P.IsPrime
hqm : Q.IsPrime
hg0 : map (algebraMap S T) P ≠ ⊥
hfg : map (algebraMap R T) p ≠ ⊥
hg : map (algebraMap S T) P ≤ Q
hf0 : map f p ≠ ⊥
hp0 : P ≠ ⊥
hq0 : Q ≠ ⊥
this : P.IsMaximal := Ring.DimensionLEOne.maximalOfPrime hp0 hpm
⊢ Multiset.count Q (normalizedFactors (map (algebraMap S T) (map f p))) =
    Multiset.count P (normalizedFactors (map f p)) * Multiset.count Q (normalizedFactors (map (algebraMap S T) P))

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : CommRing T
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
inst✝¹ : IsDedekindDomain S
inst✝ : IsDedekindDomain T
p : Ideal R
P : Ideal S
Q : Ideal T
hpm : P.IsPrime
hqm : Q.IsPrime
hg0 : map (algebraMap S T) P ≠ ⊥
hfg : map (algebraMap R T) p ≠ ⊥
hg : map (algebraMap S T) P ≤ Q
hf0 : map f p ≠ ⊥
hp0 : P ≠ ⊥
hq0 : Q ≠ ⊥
this : P.IsMaximal := Ring.DimensionLEOne.maximalOfPrime hp0 hpm
⊢ Multiset.count Q (normalizedFactors (map (algebraMap S T) (map f p))) =
    Multiset.count P (normalizedFactors (map f p)) * Multiset.count Q (normalizedFactors (map (algebraMap S T) P))
Module: Mathlib.NumberTheory.RamificationInertia.Ramification

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : CommRing T
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
inst✝¹ : IsDedekindDomain S
inst✝ : IsDedekindDomain T
p : Ideal R
P : Ideal S
Q : Ideal T
hpm : P.IsPrime
hqm : Q.IsPrime
hg0 : map (algebraMap S T) P ≠ ⊥
hfg : map (algebraMap R T) p ≠ ⊥
hg : map (algebraMap S T) P ≤ Q
hf0 : map f p ≠ ⊥
hp0 : P ≠ ⊥
hq0 : Q ≠ ⊥
this : P.IsMaximal := Ring.DimensionLEOne.maximalOfPrime hp0 hpm
I : Ideal S
hcp : P ⊔ I = ⊤
heq : map f p = P ^ Multiset.count P (normalizedFactors (map f p)) * I
⊢ ⊤ = ⊤

Tactic:
]

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

Goal state before:
pp: R : Type v
inst✝⁶ : CommRing R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : Small.{w, u} A
H :
  ∀ ⦃B : Type w⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp
B : Type u
inst✝² : CommRing B
inst✝¹ : Small.{w, u} B
inst✝ : Algebra R B
I : Ideal B
⊢ I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp

Tactic:
hI

Goal state after:
R : Type v
inst✝⁶ : CommRing R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : Small.{w, u} A
H :
  ∀ ⦃B : Type w⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp
B : Type u
inst✝² : CommRing B
inst✝¹ : Small.{w, u} B
inst✝ : Algebra R B
I : Ideal B
hI : I ^ 2 = ⊥
⊢ Function.Injective (Ideal.Quotient.mkₐ R I).comp
Module: Mathlib.RingTheory.Unramified.Basic

Goal state before:
pp: R : Type v
inst✝⁶ : CommRing R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : Small.{w, u} A
H :
  ∀ ⦃B : Type w⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp
B : Type u
inst✝² : CommRing B
inst✝¹ : Small.{w, u} B
inst✝ : Algebra R B
I : Ideal B
hI : I ^ 2 = ⊥
f g : A →ₐ[R] B
e : (Ideal.Quotient.mkₐ R I).comp f = (Ideal.Quotient.mkₐ R I).comp g
⊢ Ideal.map (Shrink.ringEquiv B).symm (I ^ 2) = ⊥

Tactic:
hI

Goal state after:
R : Type v
inst✝⁶ : CommRing R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : Small.{w, u} A
H :
  ∀ ⦃B : Type w⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp
B : Type u
inst✝² : CommRing B
inst✝¹ : Small.{w, u} B
inst✝ : Algebra R B
I : Ideal B
hI : I ^ 2 = ⊥
f g : A →ₐ[R] B
e : (Ideal.Quotient.mkₐ R I).comp f = (Ideal.Quotient.mkₐ R I).comp g
⊢ Ideal.map (Shrink.ringEquiv B).symm ⊥ = ⊥
Module: Mathlib.RingTheory.Unramified.Basic

Goal state before:
pp: R : Type v
inst✝⁶ : CommRing R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : Small.{w, u} A
H :
  ∀ ⦃B : Type w⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp
B : Type u
inst✝² : CommRing B
inst✝¹ : Small.{w, u} B
inst✝ : Algebra R B
I : Ideal B
hI : I ^ 2 = ⊥
f g : A →ₐ[R] B
e : (Ideal.Quotient.mkₐ R I).comp f = (Ideal.Quotient.mkₐ R I).comp g
⊢ Ideal.map (Shrink.ringEquiv B).symm ⊥ = ⊥

Tactic:
]

Goal state after:
R : Type v
inst✝⁶ : CommRing R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
inst✝³ : Small.{w, u} A
H :
  ∀ ⦃B : Type w⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp
B : Type u
inst✝² : CommRing B
inst✝¹ : Small.{w, u} B
inst✝ : Algebra R B
I : Ideal B
hI : I ^ 2 = ⊥
f g : A →ₐ[R] B
e : (Ideal.Quotient.mkₐ R I).comp f = (Ideal.Quotient.mkₐ R I).comp g
⊢ Ideal.map (Shrink.ringEquiv B).symm ⊥ = ⊥
Module: Mathlib.RingTheory.Unramified.Basic

Goal state before:
pp: case mpr.refine_1
R : Type v
inst✝³ : CommRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : Small.{w, u} A
H :
  ∀ ⦃B : Type u⦄ [inst : CommRing B] [Small.{w, u} B] [inst_2 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp
h : Nontrivial Ω[A⁄R]
f₁ f₂ : { f // (TensorProduct.lmul' R).kerSquareLift.comp f = AlgHom.id R A }
e : f₁ ≠ f₂
f : Ideal (A ⊗[R] A ⧸ RingHom.ker (TensorProduct.lmul' R).toRingHom ^ 2) :=
  RingHom.ker (TensorProduct.lmul' R).kerSquareLift.toRingHom
⊢ (RingHom.ker (TensorProduct.lmul' R).toRingHom).cotangentIdeal ^ 2 = ⊥

Tactic:
]

Goal state after:
case mpr.refine_1
R : Type v
inst✝³ : CommRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : Small.{w, u} A
H :
  ∀ ⦃B : Type u⦄ [inst : CommRing B] [Small.{w, u} B] [inst_2 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp
h : Nontrivial Ω[A⁄R]
f₁ f₂ : { f // (TensorProduct.lmul' R).kerSquareLift.comp f = AlgHom.id R A }
e : f₁ ≠ f₂
f : Ideal (A ⊗[R] A ⧸ RingHom.ker (TensorProduct.lmul' R).toRingHom ^ 2) :=
  RingHom.ker (TensorProduct.lmul' R).kerSquareLift.toRingHom
⊢ (RingHom.ker (TensorProduct.lmul' R).toRingHom).cotangentIdeal ^ 2 = ⊥
Module: Mathlib.RingTheory.Unramified.Basic

Goal state before:
pp: case h₁
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 : FormallyUnramified R A
I✝ : Ideal B✝
hI : IsNilpotent I✝
B : Type w
inst✝ : CommRing B
I : Ideal B
⊢ I ^ 2 = ⊥ → ∀ [inst : Algebra R B], Function.Injective (Ideal.Quotient.mkₐ R I).comp

Tactic:
hI

Goal state after:
case h₁
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 : FormallyUnramified R A
I✝ : Ideal B✝
hI✝ : IsNilpotent I✝
B : Type w
inst✝ : CommRing B
I : Ideal B
hI : I ^ 2 = ⊥
⊢ ∀ [inst : Algebra R B], Function.Injective (Ideal.Quotient.mkₐ R I).comp
Module: Mathlib.RingTheory.Unramified.Basic

Goal state before:
pp: case h₂
R : Type v
inst✝⁶ : CommRing R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
B✝ : Type w
inst✝³ : CommRing B✝
inst✝² : FormallyUnramified R A
I✝ : Ideal B✝
hI : IsNilpotent I✝
B : Type w
inst✝¹ : CommRing B
I J : Ideal B
hIJ : I ≤ J
h₁ : ∀ [inst : Algebra R B], Function.Injective (Ideal.Quotient.mkₐ R I).comp
h₂ : ∀ [inst : Algebra R (B ⧸ I)], Function.Injective (Ideal.Quotient.mkₐ R (Ideal.map (Ideal.Quotient.mk I) J)).comp
inst✝ : Algebra R B
g₁ g₂ : A →ₐ[R] B
x : A
e : (Ideal.Quotient.mk J) (g₁ x) = (Ideal.Quotient.mk J) (g₂ x)
⊢ (Ideal.Quotient.mk J) (g₁ x) = (Ideal.Quotient.mk J) (g₂ x)

Tactic:
]

Goal state after:
case h₂
R : Type v
inst✝⁶ : CommRing R
A : Type u
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
B✝ : Type w
inst✝³ : CommRing B✝
inst✝² : FormallyUnramified R A
I✝ : Ideal B✝
hI : IsNilpotent I✝
B : Type w
inst✝¹ : CommRing B
I J : Ideal B
hIJ : I ≤ J
h₁ : ∀ [inst : Algebra R B], Function.Injective (Ideal.Quotient.mkₐ R I).comp
h₂ : ∀ [inst : Algebra R (B ⧸ I)], Function.Injective (Ideal.Quotient.mkₐ R (Ideal.map (Ideal.Quotient.mk I) J)).comp
inst✝ : Algebra R B
g₁ g₂ : A →ₐ[R] B
x : A
e : (Ideal.Quotient.mk J) (g₁ x) = (Ideal.Quotient.mk J) (g₂ x)
⊢ (Ideal.Quotient.mk J) (g₁ x) = (Ideal.Quotient.mk J) (g₂ x)
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)
i : ℕ
hi : i = 0
x : A
⊢ Subsingleton (B ⧸ ⊤)

Tactic:
]

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)
i : ℕ
hi : i = 0
x : A
⊢ Subsingleton (B ⧸ ⊤)
Module: Mathlib.NumberTheory.RamificationInertia.Ramification

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : CommRing T
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
inst✝¹ : IsDedekindDomain S
inst✝ : IsDedekindDomain T
p : Ideal R
P : Ideal S
Q : Ideal T
hpm : P.IsPrime
hqm : Q.IsPrime
hg0 : map (algebraMap S T) P ≠ ⊥
hfg : map (algebraMap R T) p ≠ ⊥
hg : map (algebraMap S T) P ≤ Q
hf0 : map f p ≠ ⊥
hp0 : P ≠ ⊥
hq0 : Q ≠ ⊥
this : P.IsMaximal := Ring.DimensionLEOne.maximalOfPrime hp0 hpm
I : Ideal S
hcp✝ : P ⊔ I = ⊤
heq : map f p = P ^ Multiset.count P (normalizedFactors (map f p)) * I
hcp : ⊤ = map (algebraMap S T) P ⊔ map (algebraMap S T) I
hntq : ¬⊤ ≤ Q
⊢ Multiset.count P (normalizedFactors (map f p)) * Multiset.count Q (normalizedFactors (map (algebraMap S T) P)) +
      Multiset.count Q (normalizedFactors (map (algebraMap S T) I)) =
    Multiset.count P (normalizedFactors (map f p)) * Multiset.count Q (normalizedFactors (map (algebraMap S T) P))

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : CommRing T
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra S T
inst✝³ : Algebra R T
inst✝² : IsScalarTower R S T
inst✝¹ : IsDedekindDomain S
inst✝ : IsDedekindDomain T
p : Ideal R
P : Ideal S
Q : Ideal T
hpm : P.IsPrime
hqm : Q.IsPrime
hg0 : map (algebraMap S T) P ≠ ⊥
hfg : map (algebraMap R T) p ≠ ⊥
hg : map (algebraMap S T) P ≤ Q
hf0 : map f p ≠ ⊥
hp0 : P ≠ ⊥
hq0 : Q ≠ ⊥
this : P.IsMaximal := Ring.DimensionLEOne.maximalOfPrime hp0 hpm
I : Ideal S
hcp✝ : P ⊔ I = ⊤
heq : map f p = P ^ Multiset.count P (normalizedFactors (map f p)) * I
hcp : ⊤ = map (algebraMap S T) P ⊔ map (algebraMap S T) I
hntq : ¬⊤ ≤ Q
⊢ Multiset.count P (normalizedFactors (map f p)) * Multiset.count Q (normalizedFactors (map (algebraMap S T) P)) +
      Multiset.count Q (normalizedFactors (map (algebraMap S T) I)) =
    Multiset.count P (normalizedFactors (map f p)) * Multiset.count Q (normalizedFactors (map (algebraMap S T) P))
Module: Mathlib.RingTheory.Unramified.Basic

Goal state before:
pp: case neg
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)
i : ℕ
hi : ¬i = 0
x : A
⊢ (Ideal.Quotient.mk I) (g₁ x) = (Ideal.Quotient.mk I) (g₂ x)

pp: case neg
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)
i : ℕ
hi : ¬i = 0
x : A
⊢ RingHom.ker (Ideal.Quotient.mk (I ^ i)) ≤ I

pp: case neg.hf
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)
i : ℕ
hi : ¬i = 0
x : A
⊢ Function.Surjective ⇑(Ideal.Quotient.mk (I ^ i))

Tactic:
]

Goal state after:
case neg
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)
i : ℕ
hi : ¬i = 0
x : A
⊢ (Ideal.Quotient.mk I) (g₁ x) = (Ideal.Quotient.mk I) (g₂ x)

case neg
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)
i : ℕ
hi : ¬i = 0
x : A
⊢ RingHom.ker (Ideal.Quotient.mk (I ^ i)) ≤ I

case neg.hf
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)
i : ℕ
hi : ¬i = 0
x : A
⊢ Function.Surjective ⇑(Ideal.Quotient.mk (I ^ i))
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
⊢ ∀ (i : ℕ), g₁ x - g₂ x ∈ I ^ i

Tactic:
]

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
⊢ ∀ (i : ℕ), g₁ x - g₂ x ∈ I ^ i
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
i : ℕ
⊢ (Ideal.Quotient.mk (I ^ i)) (g₁ x) = (Ideal.Quotient.mk (I ^ i)) (g₂ x)

Tactic:
]

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
i : ℕ
⊢ (Ideal.Quotient.mk (I ^ i)) (g₁ x) = (Ideal.Quotient.mk (I ^ i)) (g₂ x)
Module: Mathlib.RingTheory.Unramified.Basic

Goal state before:
pp: R : Type u
inst✝ : CommRing R
⊢ ∀ ⦃B : Type u⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp

Tactic:
]

Goal state after:
R : Type u
inst✝ : CommRing R
⊢ ∀ ⦃B : Type u⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp