mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: case refine_2.zero
p : ℕ
hp : Fact (Nat.Prime p)
i : ℕ
hi : i < ((cyclotomic p ℤ).comp (X + 1)).natDegree
⊢ ((cyclotomic p ℤ).comp (X + 1)).coeff i ∈ ℤ ∙ ↑p

Tactic:
]

Goal state after:
case refine_2.zero
p : ℕ
hp : Fact (Nat.Prime p)
i : ℕ
hi : i < ((cyclotomic p ℤ).comp (X + 1)).natDegree
⊢ ((cyclotomic p ℤ).comp (X + 1)).coeff i ∈ ℤ ∙ ↑p
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝¹⁰ : CommRing R
inst✝⁹ : IsDomain R
S : Type u_3
inst✝⁸ : CommRing S
inst✝⁷ : IsDomain S
inst✝⁶ : IsIntegrallyClosed R
inst✝⁵ : IsIntegrallyClosed S
inst✝⁴ : Algebra R S
inst✝³ : Module.Finite R S
inst✝² : IsTorsionFree R S
inst✝¹ : IsDedekindDomain R
inst✝ : IsDedekindDomain S
I P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
K : Type u_1 := FractionRing R
⊢ span
      {(Algebra.intNorm (Localization.AtPrime P) (Localization P'))
          ((algebraMap Rₚ (Localization P')) (IsPrincipal.generator (map (algebraMap R Rₚ) I)))} =
    span {IsPrincipal.generator (map (algebraMap R Rₚ) I) ^ finrank R S}

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹⁰ : CommRing R
inst✝⁹ : IsDomain R
S : Type u_3
inst✝⁸ : CommRing S
inst✝⁷ : IsDomain S
inst✝⁶ : IsIntegrallyClosed R
inst✝⁵ : IsIntegrallyClosed S
inst✝⁴ : Algebra R S
inst✝³ : Module.Finite R S
inst✝² : IsTorsionFree R S
inst✝¹ : IsDedekindDomain R
inst✝ : IsDedekindDomain S
I P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
K : Type u_1 := FractionRing R
⊢ span
      {(Algebra.intNorm (Localization.AtPrime P) (Localization P'))
          ((algebraMap Rₚ (Localization P')) (IsPrincipal.generator (map (algebraMap R Rₚ) I)))} =
    span {IsPrincipal.generator (map (algebraMap R Rₚ) I) ^ finrank R S}
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝¹⁰ : CommRing R
inst✝⁹ : IsDomain R
S : Type u_3
inst✝⁸ : CommRing S
inst✝⁷ : IsDomain S
inst✝⁶ : IsIntegrallyClosed R
inst✝⁵ : IsIntegrallyClosed S
inst✝⁴ : Algebra R S
inst✝³ : Module.Finite R S
inst✝² : IsTorsionFree R S
inst✝¹ : IsDedekindDomain R
inst✝ : IsDedekindDomain S
I P : Ideal R
hPd : P.IsMaximal
P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl
Rₚ : Type u_1 := Localization.AtPrime P
K : Type u_1 := FractionRing R
⊢ (algebraMap Rₚ K) (IsPrincipal.generator (map (algebraMap R Rₚ) I)) ^ finrank R S =
    (algebraMap Rₚ K) (IsPrincipal.generator (map (algebraMap R Rₚ) I)) ^ finrank R S

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝¹⁴ : CommRing R
inst✝¹³ : IsDomain R
S : Type u_3
inst✝¹² : CommRing S
inst✝¹¹ : IsDomain S
inst✝¹⁰ : IsIntegrallyClosed R
inst✝⁹ : IsIntegrallyClosed S
inst✝⁸ : Algebra R S
inst✝⁷ : Module.Finite R S
inst✝⁶ : IsTorsionFree R S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : IsDedekindDomain S
R' : Type u_4
inst✝³ : CommRing R'
I : Ideal R'
inst✝² : Algebra R' R
inst✝¹ : Algebra R' S
inst✝ : IsScalarTower R' R S
⊢ (relNorm R) (map ((algebraMap R S).comp (algebraMap R' R)) I) =
    (relNorm R) (map ((algebraMap R S).comp (algebraMap R' R)) I)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: case pos
R : Type u_1
inst✝¹¹ : CommRing R
inst✝¹⁰ : IsDomain R
S : Type u_3
inst✝⁹ : CommRing S
inst✝⁸ : IsDomain S
inst✝⁷ : IsIntegrallyClosed R
inst✝⁶ : IsIntegrallyClosed S
inst✝⁵ : Algebra R S
inst✝⁴ : Module.Finite R S
inst✝³ : IsTorsionFree R S
inst✝² : IsDedekindDomain R
inst✝¹ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝ : p.IsPrime
hp : p = ⊥
this : P.LiesOver ⊥
⊢ ⊥ = ⊥

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: case neg
R : Type u_1
inst✝¹¹ : CommRing R
inst✝¹⁰ : IsDomain R
S : Type u_3
inst✝⁹ : CommRing S
inst✝⁸ : IsDomain S
inst✝⁷ : IsIntegrallyClosed R
inst✝⁶ : IsIntegrallyClosed S
inst✝⁵ : Algebra R S
inst✝⁴ : Module.Finite R S
inst✝³ : IsTorsionFree R S
inst✝² : IsDedekindDomain R
inst✝¹ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝ : p.IsPrime
hp : ¬p = ⊥
h : ∃ i ≤ finrank R S, Associated ((relNorm R) P) (p ^ i)
⊢ ∃ s, (relNorm R) P = p ^ s

Tactic:
]

Goal state after:
case neg
R : Type u_1
inst✝¹¹ : CommRing R
inst✝¹⁰ : IsDomain R
S : Type u_3
inst✝⁹ : CommRing S
inst✝⁸ : IsDomain S
inst✝⁷ : IsIntegrallyClosed R
inst✝⁶ : IsIntegrallyClosed S
inst✝⁵ : Algebra R S
inst✝⁴ : Module.Finite R S
inst✝³ : IsTorsionFree R S
inst✝² : IsDedekindDomain R
inst✝¹ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝ : p.IsPrime
hp : ¬p = ⊥
h : ∃ i ≤ finrank R S, Associated ((relNorm R) P) (p ^ i)
⊢ ∃ s, (relNorm R) P = p ^ s
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝¹¹ : CommRing R
inst✝¹⁰ : IsDomain R
S : Type u_3
inst✝⁹ : CommRing S
inst✝⁸ : IsDomain S
inst✝⁷ : IsIntegrallyClosed R
inst✝⁶ : IsIntegrallyClosed S
inst✝⁵ : Algebra R S
inst✝⁴ : Module.Finite R S
inst✝³ : IsTorsionFree R S
inst✝² : IsDedekindDomain R
inst✝¹ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝ : p.IsPrime
hp : ¬p = ⊥
s : ℕ
left✝ : s ≤ finrank R S
hs : (relNorm R) P = p ^ s
⊢ (relNorm R) P = p ^ s

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹¹ : CommRing R
inst✝¹⁰ : IsDomain R
S : Type u_3
inst✝⁹ : CommRing S
inst✝⁸ : IsDomain S
inst✝⁷ : IsIntegrallyClosed R
inst✝⁶ : IsIntegrallyClosed S
inst✝⁵ : Algebra R S
inst✝⁴ : Module.Finite R S
inst✝³ : IsTorsionFree R S
inst✝² : IsDedekindDomain R
inst✝¹ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝ : p.IsPrime
hp : ¬p = ⊥
s : ℕ
left✝ : s ≤ finrank R S
hs : (relNorm R) P = p ^ s
⊢ (relNorm R) P = p ^ s
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: case refine_2.succ
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
hn :
  ∀ {n_1 : ℕ},
    n_1 < ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).natDegree →
      ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).coeff n_1 ∈ ℤ ∙ ↑p
i : ℕ
hi : i < ((cyclotomic (p ^ (n + 1 + 1)) ℤ).comp (X + 1)).natDegree
⊢ (if p ∣ i then ((cyclotomic (p ^ (n + 1)) (ZMod p)).comp (X + 1)).coeff (i / p) else 0) = 0

pp: case refine_2.succ.hn
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
hn :
  ∀ {n_1 : ℕ},
    n_1 < ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).natDegree →
      ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).coeff n_1 ∈ ℤ ∙ ↑p
i : ℕ
hi : i < ((cyclotomic (p ^ (n + 1 + 1)) ℤ).comp (X + 1)).natDegree
⊢ p ∣ p ^ (n + 1)

Tactic:
]

Goal state after:
case refine_2.succ
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
hn :
  ∀ {n_1 : ℕ},
    n_1 < ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).natDegree →
      ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).coeff n_1 ∈ ℤ ∙ ↑p
i : ℕ
hi : i < ((cyclotomic (p ^ (n + 1 + 1)) ℤ).comp (X + 1)).natDegree
⊢ (if p ∣ i then ((cyclotomic (p ^ (n + 1)) (ZMod p)).comp (X + 1)).coeff (i / p) else 0) = 0

case refine_2.succ.hn
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
hn :
  ∀ {n_1 : ℕ},
    n_1 < ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).natDegree →
      ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).coeff n_1 ∈ ℤ ∙ ↑p
i : ℕ
hi : i < ((cyclotomic (p ^ (n + 1 + 1)) ℤ).comp (X + 1)).natDegree
⊢ p ∣ p ^ (n + 1)
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDomain R
S : Type u_3
inst✝¹¹ : CommRing S
inst✝¹⁰ : IsDomain S
inst✝⁹ : IsIntegrallyClosed R
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : Algebra R S
inst✝⁶ : Module.Finite R S
inst✝⁵ : IsTorsionFree R S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver ⊥
inst✝² : p.IsMaximal
inst✝¹ : P.IsPrime
inst✝ : IsGalois (FractionRing R) (FractionRing S)
this✝¹ : P.IsMaximal
G : Type u_3 := Gal(FractionRing S/FractionRing R)
this✝ : MulSemiringAction Gal(FractionRing S/FractionRing R) S :=
  IsIntegralClosure.MulSemiringAction R (FractionRing R) (FractionRing S) S
this : IsGaloisGroup G R S
hp : p = ⊥
h : P.inertiaDeg R ≠ 0
⊢ P = ⊥

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDomain R
S : Type u_3
inst✝¹¹ : CommRing S
inst✝¹⁰ : IsDomain S
inst✝⁹ : IsIntegrallyClosed R
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : Algebra R S
inst✝⁶ : Module.Finite R S
inst✝⁵ : IsTorsionFree R S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver ⊥
inst✝² : p.IsMaximal
inst✝¹ : P.IsPrime
inst✝ : IsGalois (FractionRing R) (FractionRing S)
this✝¹ : P.IsMaximal
G : Type u_3 := Gal(FractionRing S/FractionRing R)
this✝ : MulSemiringAction Gal(FractionRing S/FractionRing R) S :=
  IsIntegralClosure.MulSemiringAction R (FractionRing R) (FractionRing S) S
this : IsGaloisGroup G R S
hp : p = ⊥
h : P.inertiaDeg R ≠ 0
⊢ P = ⊥
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: case refine_2.succ
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
hn : ∀ {n_1 : ℕ}, n_1 < p ^ n * (p - 1) → ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + C 1)).coeff n_1 ∈ ℤ ∙ ↑p
i : ℕ
hi : i < p ^ (n + 1) * (p - 1)
k : ℕ
hk : i = p * k
⊢ ((cyclotomic (p ^ (n + 1)) (ZMod p)).comp (X + 1)).coeff (i / p) = 0

Tactic:
]

Goal state after:
case refine_2.succ
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
hn : ∀ {n_1 : ℕ}, n_1 < p ^ n * (p - 1) → ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + C 1)).coeff n_1 ∈ ℤ ∙ ↑p
i : ℕ
hi : i < p ^ (n + 1) * (p - 1)
k : ℕ
hk : i = p * k
⊢ ((cyclotomic (p ^ (n + 1)) (ZMod p)).comp (X + 1)).coeff (i / p) = 0
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: case pos
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDomain R
S : Type u_3
inst✝¹¹ : CommRing S
inst✝¹⁰ : IsDomain S
inst✝⁹ : IsIntegrallyClosed R
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : Algebra R S
inst✝⁶ : Module.Finite R S
inst✝⁵ : IsTorsionFree R S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝² : p.IsMaximal
inst✝¹ : P.IsPrime
inst✝ : IsGalois (FractionRing R) (FractionRing S)
this✝¹ : P.IsMaximal
G : Type u_3 := Gal(FractionRing S/FractionRing R)
this✝ : MulSemiringAction Gal(FractionRing S/FractionRing R) S :=
  IsIntegralClosure.MulSemiringAction R (FractionRing R) (FractionRing S) S
this : IsGaloisGroup G R S
hp : p = ⊥
h : P.inertiaDeg R ≠ 0
hP : P = ⊥
⊢ ⊥ = ⊥

pp: case pos
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDomain R
S : Type u_3
inst✝¹¹ : CommRing S
inst✝¹⁰ : IsDomain S
inst✝⁹ : IsIntegrallyClosed R
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : Algebra R S
inst✝⁶ : Module.Finite R S
inst✝⁵ : IsTorsionFree R S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝² : p.IsMaximal
inst✝¹ : P.IsPrime
inst✝ : IsGalois (FractionRing R) (FractionRing S)
this✝¹ : P.IsMaximal
G : Type u_3 := Gal(FractionRing S/FractionRing R)
this✝ : MulSemiringAction Gal(FractionRing S/FractionRing R) S :=
  IsIntegralClosure.MulSemiringAction R (FractionRing R) (FractionRing S) S
this : IsGaloisGroup G R S
hp : p = ⊥
h : P.inertiaDeg R ≠ 0
hP : P = ⊥
⊢ ⊥.inertiaDeg R ≠ 0

Tactic:
]

Goal state after:
case pos
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDomain R
S : Type u_3
inst✝¹¹ : CommRing S
inst✝¹⁰ : IsDomain S
inst✝⁹ : IsIntegrallyClosed R
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : Algebra R S
inst✝⁶ : Module.Finite R S
inst✝⁵ : IsTorsionFree R S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝² : p.IsMaximal
inst✝¹ : P.IsPrime
inst✝ : IsGalois (FractionRing R) (FractionRing S)
this✝¹ : P.IsMaximal
G : Type u_3 := Gal(FractionRing S/FractionRing R)
this✝ : MulSemiringAction Gal(FractionRing S/FractionRing R) S :=
  IsIntegralClosure.MulSemiringAction R (FractionRing R) (FractionRing S) S
this : IsGaloisGroup G R S
hp : p = ⊥
h : P.inertiaDeg R ≠ 0
hP : P = ⊥
⊢ ⊥.inertiaDeg R ≠ 0
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: case pos
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDomain R
S : Type u_3
inst✝¹¹ : CommRing S
inst✝¹⁰ : IsDomain S
inst✝⁹ : IsIntegrallyClosed R
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : Algebra R S
inst✝⁶ : Module.Finite R S
inst✝⁵ : IsTorsionFree R S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝² : p.IsMaximal
inst✝¹ : P.IsPrime
inst✝ : IsGalois (FractionRing R) (FractionRing S)
this✝¹ : P.IsMaximal
G : Type u_3 := Gal(FractionRing S/FractionRing R)
this✝ : MulSemiringAction Gal(FractionRing S/FractionRing R) S :=
  IsIntegralClosure.MulSemiringAction R (FractionRing R) (FractionRing S) S
this : IsGaloisGroup G R S
hp : p = ⊥
h : ⊥.inertiaDeg R ≠ 0
hP : P = ⊥
⊢ ⊥.inertiaDeg R ≠ 0

Tactic:
]

Goal state after:
case pos
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDomain R
S : Type u_3
inst✝¹¹ : CommRing S
inst✝¹⁰ : IsDomain S
inst✝⁹ : IsIntegrallyClosed R
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : Algebra R S
inst✝⁶ : Module.Finite R S
inst✝⁵ : IsTorsionFree R S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝² : p.IsMaximal
inst✝¹ : P.IsPrime
inst✝ : IsGalois (FractionRing R) (FractionRing S)
this✝¹ : P.IsMaximal
G : Type u_3 := Gal(FractionRing S/FractionRing R)
this✝ : MulSemiringAction Gal(FractionRing S/FractionRing R) S :=
  IsIntegralClosure.MulSemiringAction R (FractionRing R) (FractionRing S) S
this : IsGaloisGroup G R S
hp : p = ⊥
h : ⊥.inertiaDeg R ≠ 0
hP : P = ⊥
⊢ ⊥.inertiaDeg R ≠ 0
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: case refine_2.succ
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
hn : ∀ {n_1 : ℕ}, n_1 < p ^ n * (p - 1) → ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + C 1)).coeff n_1 ∈ ℤ ∙ ↑p
i k : ℕ
hi : p * k < p * (p ^ n * (p - 1))
hk : i = p * k
⊢ ((cyclotomic (p ^ (n + 1)) (ZMod p)).comp (X + 1)).coeff (i / p) = 0

Tactic:
]

Goal state after:
case refine_2.succ
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
hn : ∀ {n_1 : ℕ}, n_1 < p ^ n * (p - 1) → ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + C 1)).coeff n_1 ∈ ℤ ∙ ↑p
i k : ℕ
hi : p * k < p * (p ^ n * (p - 1))
hk : i = p * k
⊢ ((cyclotomic (p ^ (n + 1)) (ZMod p)).comp (X + 1)).coeff (i / p) = 0
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDomain R
S : Type u_3
inst✝¹¹ : CommRing S
inst✝¹⁰ : IsDomain S
inst✝⁹ : IsIntegrallyClosed R
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : Algebra R S
inst✝⁶ : Module.Finite R S
inst✝⁵ : IsTorsionFree R S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝² : p.IsMaximal
inst✝¹ : P.IsPrime
inst✝ : IsGalois (FractionRing R) (FractionRing S)
this✝² : P.IsMaximal
G : Type u_3 := Gal(FractionRing S/FractionRing R)
this✝¹ : MulSemiringAction Gal(FractionRing S/FractionRing R) S :=
  IsIntegralClosure.MulSemiringAction R (FractionRing R) (FractionRing S) S
this✝ : IsGaloisGroup G R S
hp : ¬p = ⊥
s : ℕ
hs : (relNorm R) P = p ^ P.inertiaDeg R
this : s = P.inertiaDeg R
⊢ (relNorm R) P = p ^ P.inertiaDeg R

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDomain R
S : Type u_3
inst✝¹¹ : CommRing S
inst✝¹⁰ : IsDomain S
inst✝⁹ : IsIntegrallyClosed R
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : Algebra R S
inst✝⁶ : Module.Finite R S
inst✝⁵ : IsTorsionFree R S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝² : p.IsMaximal
inst✝¹ : P.IsPrime
inst✝ : IsGalois (FractionRing R) (FractionRing S)
this✝² : P.IsMaximal
G : Type u_3 := Gal(FractionRing S/FractionRing R)
this✝¹ : MulSemiringAction Gal(FractionRing S/FractionRing R) S :=
  IsIntegralClosure.MulSemiringAction R (FractionRing R) (FractionRing S) S
this✝ : IsGaloisGroup G R S
hp : ¬p = ⊥
s : ℕ
hs : (relNorm R) P = p ^ P.inertiaDeg R
this : s = P.inertiaDeg R
⊢ (relNorm R) P = p ^ P.inertiaDeg R
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: case refine_2.succ
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
hn : ∀ {n_1 : ℕ}, n_1 < p ^ n * (p - 1) → ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + C 1)).coeff n_1 ∈ ℤ ∙ ↑p
i k : ℕ
hi : p * k < p * (p ^ n * (p - 1))
hk : i = p * k
⊢ ((cyclotomic (p ^ (n + 1)) (ZMod p)).comp (X + 1)).coeff k = 0

Tactic:
]

Goal state after:
case refine_2.succ
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
hn : ∀ {n_1 : ℕ}, n_1 < p ^ n * (p - 1) → ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + C 1)).coeff n_1 ∈ ℤ ∙ ↑p
i k : ℕ
hi : p * k < p * (p ^ n * (p - 1))
hk : i = p * k
⊢ ((cyclotomic (p ^ (n + 1)) (ZMod p)).comp (X + 1)).coeff k = 0
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: case refine_2.succ
p : ℕ
hp : Fact (Nat.Prime p)
n i k : ℕ
hi : p * k < p * (p ^ n * (p - 1))
hk : i = p * k
hn : (Polynomial.map (Int.castRingHom (ZMod p)) ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + C 1))).coeff k = 0
⊢ ((cyclotomic (p ^ (n + 1)) (ZMod p)).comp (X + 1)).coeff k = 0

Tactic:
]

Goal state after:
case refine_2.succ
p : ℕ
hp : Fact (Nat.Prime p)
n i k : ℕ
hi : p * k < p * (p ^ n * (p - 1))
hk : i = p * k
hn : (Polynomial.map (Int.castRingHom (ZMod p)) ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + C 1))).coeff k = 0
⊢ ((cyclotomic (p ^ (n + 1)) (ZMod p)).comp (X + 1)).coeff k = 0
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDomain R
S : Type u_3
inst✝¹¹ : CommRing S
inst✝¹⁰ : IsDomain S
inst✝⁹ : IsIntegrallyClosed R
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : Algebra R S
inst✝⁶ : Module.Finite R S
inst✝⁵ : IsTorsionFree R S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝² : p.IsMaximal
inst✝¹ : P.IsPrime
inst✝ : IsGalois (FractionRing R) (FractionRing S)
this✝¹ : P.IsMaximal
G : Type u_3 := Gal(FractionRing S/FractionRing R)
this✝ : MulSemiringAction Gal(FractionRing S/FractionRing R) S :=
  IsIntegralClosure.MulSemiringAction R (FractionRing R) (FractionRing S) S
this : IsGaloisGroup G R S
hp : ¬p = ⊥
s : ℕ
hs : (relNorm R) P = p ^ s
Q : Ideal S
hQ : Q ∈ p.primesOver S
⊢ (relNorm R) Q ^ Q.ramificationIdx R = p ^ (p.ramificationIdxIn S * s)

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDomain R
S : Type u_3
inst✝¹¹ : CommRing S
inst✝¹⁰ : IsDomain S
inst✝⁹ : IsIntegrallyClosed R
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : Algebra R S
inst✝⁶ : Module.Finite R S
inst✝⁵ : IsTorsionFree R S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝² : p.IsMaximal
inst✝¹ : P.IsPrime
inst✝ : IsGalois (FractionRing R) (FractionRing S)
this✝¹ : P.IsMaximal
G : Type u_3 := Gal(FractionRing S/FractionRing R)
this✝ : MulSemiringAction Gal(FractionRing S/FractionRing R) S :=
  IsIntegralClosure.MulSemiringAction R (FractionRing R) (FractionRing S) S
this : IsGaloisGroup G R S
hp : ¬p = ⊥
s : ℕ
hs : (relNorm R) P = p ^ s
Q : Ideal S
hQ : Q ∈ p.primesOver S
⊢ (relNorm R) Q ^ Q.ramificationIdx R = p ^ (p.ramificationIdxIn S * s)
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDomain R
S : Type u_3
inst✝¹¹ : CommRing S
inst✝¹⁰ : IsDomain S
inst✝⁹ : IsIntegrallyClosed R
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : Algebra R S
inst✝⁶ : Module.Finite R S
inst✝⁵ : IsTorsionFree R S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝² : p.IsMaximal
inst✝¹ : P.IsPrime
inst✝ : IsGalois (FractionRing R) (FractionRing S)
this✝³ : P.IsMaximal
G : Type u_3 := Gal(FractionRing S/FractionRing R)
this✝² : MulSemiringAction Gal(FractionRing S/FractionRing R) S :=
  IsIntegralClosure.MulSemiringAction R (FractionRing R) (FractionRing S) S
this✝¹ : IsGaloisGroup G R S
hp : ¬p = ⊥
s : ℕ
hs : (relNorm R) P = p ^ s
Q : Ideal S
hQ : Q ∈ p.primesOver S
this✝ : Q.IsPrime
this : Q.LiesOver p
⊢ (relNorm R) Q ^ p.ramificationIdxIn S = p ^ (p.ramificationIdxIn S * s)

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDomain R
S : Type u_3
inst✝¹¹ : CommRing S
inst✝¹⁰ : IsDomain S
inst✝⁹ : IsIntegrallyClosed R
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : Algebra R S
inst✝⁶ : Module.Finite R S
inst✝⁵ : IsTorsionFree R S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝² : p.IsMaximal
inst✝¹ : P.IsPrime
inst✝ : IsGalois (FractionRing R) (FractionRing S)
this✝³ : P.IsMaximal
G : Type u_3 := Gal(FractionRing S/FractionRing R)
this✝² : MulSemiringAction Gal(FractionRing S/FractionRing R) S :=
  IsIntegralClosure.MulSemiringAction R (FractionRing R) (FractionRing S) S
this✝¹ : IsGaloisGroup G R S
hp : ¬p = ⊥
s : ℕ
hs : (relNorm R) P = p ^ s
Q : Ideal S
hQ : Q ∈ p.primesOver S
this✝ : Q.IsPrime
this : Q.LiesOver p
⊢ (relNorm R) Q ^ p.ramificationIdxIn S = p ^ (p.ramificationIdxIn S * s)
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
hn :
  ∀ {n_1 : ℕ},
    n_1 < ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).natDegree →
      ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + 1)).coeff n_1 ∈ ℤ ∙ ↑p
i : ℕ
hi : i < ((cyclotomic (p ^ (n + 1 + 1)) ℤ).comp (X + 1)).natDegree
⊢ p * p ^ n = p * p ^ n

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDomain R
S : Type u_3
inst✝¹¹ : CommRing S
inst✝¹⁰ : IsDomain S
inst✝⁹ : IsIntegrallyClosed R
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : Algebra R S
inst✝⁶ : Module.Finite R S
inst✝⁵ : IsTorsionFree R S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝² : p.IsMaximal
inst✝¹ : P.IsPrime
inst✝ : IsGalois (FractionRing R) (FractionRing S)
this✝³ : P.IsMaximal
G : Type u_3 := Gal(FractionRing S/FractionRing R)
this✝² : MulSemiringAction Gal(FractionRing S/FractionRing R) S :=
  IsIntegralClosure.MulSemiringAction R (FractionRing R) (FractionRing S) S
this✝¹ : IsGaloisGroup G R S
hp : ¬p = ⊥
s : ℕ
hs : (relNorm R) P = p ^ s
σ : G
hQ : σ • P ∈ p.primesOver S
this✝ : (σ • P).IsPrime
this : (σ • P).LiesOver p
⊢ p ^ (p.ramificationIdxIn S * s) = p ^ (p.ramificationIdxIn S * s)

Tactic:
]

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

Goal state before:
pp: case refine_3
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ eval (0 + eval 0 1) (∑ i ∈ range p, (X ^ p ^ n) ^ i) ∉ (ℤ ∙ ↑p) ^ 2

Tactic:
eval_one,

Goal state after:
case refine_3
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ eval (0 + 1) (∑ i ∈ range p, (X ^ p ^ n) ^ i) ∉ (ℤ ∙ ↑p) ^ 2
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: case refine_3
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ ∑ i ∈ range p, eval 1 ((X ^ p ^ n) ^ i) ∉ (ℤ ∙ ↑p) ^ 2

Tactic:
]

Goal state after:
case refine_3
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ ∑ i ∈ range p, eval 1 ((X ^ p ^ n) ^ i) ∉ (ℤ ∙ ↑p) ^ 2