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

Goal state before:
pp: A : Type u_1
inst✝⁸ : CommRing A
B : Type u_2
inst✝⁷ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁶ : CommRing K
inst✝⁵ : Algebra A K
inst✝⁴ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝³ : CommRing L
inst✝² : Algebra B L
inst✝¹ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I : FractionalIdeal M K
inst✝ : IsDomain B
hf' : Function.Injective ⇑f
hI : I ≠ 0
hN : 0 ∉ N
⊢ ∃ x ∈ I, x ≠ 0

Tactic:
simpa using

Goal state after:
A : Type u_1
inst✝⁸ : CommRing A
B : Type u_2
inst✝⁷ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁶ : CommRing K
inst✝⁵ : Algebra A K
inst✝⁴ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝³ : CommRing L
inst✝² : Algebra B L
inst✝¹ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I : FractionalIdeal M K
inst✝ : IsDomain B
hf' : Function.Injective ⇑f
hI : I ≠ 0
hN : 0 ∉ N
⊢ ∃ x ∈ I, ¬x = 0
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
b : B
left✝ : b ∈ ↑⊤
⊢ 1 ∈ ↑1 ∧ (IsLocalization.map L f hf) 1 = 1

Tactic:
simpa using

Goal state after:
A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
b : B
left✝ : b ∈ ↑⊤
⊢ ∃ x', (algebraMap A K) x' = 1
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: case refine_2.refine_1
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsComplex}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this✝¹ : CharZero ↥K
this✝ : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsComplex}
this : minkowskiBound (↥K) 1 < ↑(convexBodyLT'Factor ↥K) * ↑(boundOfDiscBdd N)
x : 𝓞 ↥K
hx₁ : ℚ⟮↑x⟯ = ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), w ↑x < √(1 + ↑(boundOfDiscBdd N) ^ 2)
hx : IsIntegral ℤ ((algebraMap (𝓞 ↥K) ↥K) x)
i : ℕ
⊢ ↑|(minpoly ℤ ↑x).coeff i| = ‖(minpoly ℚ ↑x).coeff i‖

Tactic:
minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hx,

Goal state after:
case refine_2.refine_1
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsComplex}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this✝¹ : CharZero ↥K
this✝ : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsComplex}
this : minkowskiBound (↥K) 1 < ↑(convexBodyLT'Factor ↥K) * ↑(boundOfDiscBdd N)
x : 𝓞 ↥K
hx₁ : ℚ⟮↑x⟯ = ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), w ↑x < √(1 + ↑(boundOfDiscBdd N) ^ 2)
hx : IsIntegral ℤ ((algebraMap (𝓞 ↥K) ↥K) x)
i : ℕ
⊢ ↑|(minpoly ℤ ↑x).coeff i| = ‖(Polynomial.map (algebraMap ℤ ℚ) (minpoly ℤ ((algebraMap (𝓞 ↥K) ↥K) x))).coeff i‖
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: case refine_2.refine_1
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsComplex}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this✝¹ : CharZero ↥K
this✝ : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsComplex}
this : minkowskiBound (↥K) 1 < ↑(convexBodyLT'Factor ↥K) * ↑(boundOfDiscBdd N)
x : 𝓞 ↥K
hx₁ : ℚ⟮↑x⟯ = ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), w ↑x < √(1 + ↑(boundOfDiscBdd N) ^ 2)
hx : IsIntegral ℤ ((algebraMap (𝓞 ↥K) ↥K) x)
i : ℕ
⊢ ↑|(minpoly ℤ ↑x).coeff i| = ‖(minpoly ℚ ↑x).coeff i‖

Tactic:
rw [minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hx, coeff_map, eq_intCast,
          Int.norm_cast_rat, Int.norm_eq_abs, Int.cast_abs]

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

Goal state before:
pp: S : Type u_2
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Module.Free ℤ S
I : Ideal S
h✝ : Finite (S ⧸ I)
this : Fintype (S ⧸ I)
d : ℕ
h : ∀ (x : S ⧸ I), (Ideal.Quotient.mk I) ↑d * x = 0
⊢ (Ideal.Quotient.mk I) ↑d = 0

Tactic:
simpa using

Goal state after:
S : Type u_2
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Module.Free ℤ S
I : Ideal S
h✝ : Finite (S ⧸ I)
this : Fintype (S ⧸ I)
d : ℕ
h : ∀ (x : S ⧸ I), (Ideal.Quotient.mk I) ↑d * x = 0
⊢ ↑d = 0
Module: Mathlib.RingTheory.Ideal.Int

Goal state before:
pp: S : Type u_2
inst✝ : CommRing S
I : Ideal S
x : ℕ
⊢ (Ideal.Quotient.mk I) ↑x = 0 ↔ absNorm (under ℤ I) ∣ x

Tactic:
Quotient.eq_zero_iff_mem,

Goal state after:
S : Type u_2
inst✝ : CommRing S
I : Ideal S
x : ℕ
⊢ ↑x ∈ I ↔ absNorm (under ℤ I) ∣ x
Module: Mathlib.RingTheory.DedekindDomain.Instances

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 : IsDomain R
inst✝⁴ : IsDomain S
inst✝³ : IsDomain T
inst✝² : Algebra R S
P : Ideal R
inst✝¹ : P.IsPrime
inst✝ : FaithfulSMul R S
x✝ : ↥P.primeCompl
x : R
hx : x ∈ P.primeCompl
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
simpa using

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 : IsDomain R
inst✝⁴ : IsDomain S
inst✝³ : IsDomain T
inst✝² : Algebra R S
P : Ideal R
inst✝¹ : P.IsPrime
inst✝ : FaithfulSMul R S
x✝ : ↥P.primeCompl
x : R
hx : x ∈ P.primeCompl
⊢ ¬x = 0
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
S : Type u_3
inst✝⁶ : CommRing S
inst✝⁵ : IsDomain S
inst✝⁴ : IsIntegrallyClosed R
inst✝³ : IsIntegrallyClosed S
inst✝² : Algebra R S
inst✝¹ : Module.Finite R S
inst✝ : IsTorsionFree R S
x : R
hx : x ∈ ⇑(Algebra.intNorm R S) '' ↑⊥
⊢ x = 0

Tactic:
simpa using

Goal state after:
R : Type u_1
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
S : Type u_3
inst✝⁶ : CommRing S
inst✝⁵ : IsDomain S
inst✝⁴ : IsIntegrallyClosed R
inst✝³ : IsIntegrallyClosed S
inst✝² : Algebra R S
inst✝¹ : Module.Finite R S
inst✝ : IsTorsionFree R S
x : R
hx : x ∈ ⇑(Algebra.intNorm R S) '' ↑⊥
⊢ x = 0
Module: Mathlib.NumberTheory.NumberField.Discriminant.Different

Goal state before:
pp: K : Type u_1
𝒪 : Type u_2
inst✝⁷ : Field K
inst✝⁶ : NumberField K
inst✝⁵ : CommRing 𝒪
inst✝⁴ : Algebra 𝒪 K
inst✝³ : IsFractionRing 𝒪 K
inst✝² : IsDedekindDomain 𝒪
inst✝¹ : CharZero 𝒪
inst✝ : Module.Finite ℤ 𝒪
⊢ ↑(differentIdeal ℤ 𝒪) ≤ 1⁻¹

Tactic:
simpa using

Goal state after:
K : Type u_1
𝒪 : Type u_2
inst✝⁷ : Field K
inst✝⁶ : NumberField K
inst✝⁵ : CommRing 𝒪
inst✝⁴ : Algebra 𝒪 K
inst✝³ : IsFractionRing 𝒪 K
inst✝² : IsDedekindDomain 𝒪
inst✝¹ : CharZero 𝒪
inst✝ : Module.Finite ℤ 𝒪
⊢ ↑(differentIdeal ℤ 𝒪) ≤ 1
Module: Mathlib.NumberTheory.NumberField.Discriminant.Different

Goal state before:
pp: K : Type u_1
𝒪 : Type u_2
inst✝⁷ : Field K
inst✝⁶ : NumberField K
inst✝⁵ : CommRing 𝒪
inst✝⁴ : Algebra 𝒪 K
inst✝³ : IsFractionRing 𝒪 K
inst✝² : IsDedekindDomain 𝒪
inst✝¹ : CharZero 𝒪
inst✝ : Module.Finite ℤ 𝒪
this✝ :
  (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) ≃ₗ[𝒪]
    ↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1
this :
  Nat.card (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) =
    Nat.card (↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1)
b : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := integralBasis K
b' : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := (Algebra.traceForm ℚ K).dualBasis ⋯ b
x✝ : K
e : 𝓞 K ≃ₐ[ℤ] 𝒪 := IsIntegralClosure.equiv ℤ (𝓞 K) K 𝒪
⊢ x✝ ∈ Submodule.span ℤ (Set.range ⇑b) ↔ x✝ ∈ Submodule.restrictScalars ℤ 1

Tactic:
simpa using

Goal state after:
K : Type u_1
𝒪 : Type u_2
inst✝⁷ : Field K
inst✝⁶ : NumberField K
inst✝⁵ : CommRing 𝒪
inst✝⁴ : Algebra 𝒪 K
inst✝³ : IsFractionRing 𝒪 K
inst✝² : IsDedekindDomain 𝒪
inst✝¹ : CharZero 𝒪
inst✝ : Module.Finite ℤ 𝒪
this✝ :
  (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) ≃ₗ[𝒪]
    ↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1
this :
  Nat.card (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) =
    Nat.card (↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1)
b : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := integralBasis K
b' : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := (Algebra.traceForm ℚ K).dualBasis ⋯ b
x✝ : K
e : 𝓞 K ≃ₐ[ℤ] 𝒪 := IsIntegralClosure.equiv ℤ (𝓞 K) K 𝒪
⊢ x✝ ∈ Submodule.span ℤ (Set.range ⇑b) ↔ ∃ b, (algebraMap (𝓞 K) K) b = x✝
Module: Mathlib.NumberTheory.NumberField.Discriminant.Different

Goal state before:
pp: case refine_3
K : Type u_1
𝒪 : Type u_2
inst✝⁷ : Field K
inst✝⁶ : NumberField K
inst✝⁵ : CommRing 𝒪
inst✝⁴ : Algebra 𝒪 K
inst✝³ : IsFractionRing 𝒪 K
inst✝² : IsDedekindDomain 𝒪
inst✝¹ : CharZero 𝒪
inst✝ : Module.Finite ℤ 𝒪
this✝ :
  (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) ≃ₗ[𝒪]
    ↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1
this :
  Nat.card (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) =
    Nat.card (↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1)
b : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := ⋯
b' : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := ⋯
hb : Submodule.span ℤ (Set.range ⇑b) = Submodule.restrictScalars ℤ 1
⊢ AddSubgroup.toIntSubmodule (↑(FractionalIdeal.dual ℤ ℚ 1)).toAddSubgroup =
    AddSubgroup.toIntSubmodule (AddSubgroup.closure (Set.range ⇑b'))

Tactic:
rw [AddSubgroup.toIntSubmodule_closure, ← LinearMap.BilinForm.dualSubmodule_span_of_basis, hb]

Goal state after:
case refine_3
K : Type u_1
𝒪 : Type u_2
inst✝⁷ : Field K
inst✝⁶ : NumberField K
inst✝⁵ : CommRing 𝒪
inst✝⁴ : Algebra 𝒪 K
inst✝³ : IsFractionRing 𝒪 K
inst✝² : IsDedekindDomain 𝒪
inst✝¹ : CharZero 𝒪
inst✝ : Module.Finite ℤ 𝒪
this✝ :
  (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) ≃ₗ[𝒪]
    ↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1
this :
  Nat.card (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) =
    Nat.card (↥↑(↑(differentIdeal ℤ 𝒪))⁻¹ ⧸ Submodule.comap (↑(↑(differentIdeal ℤ 𝒪))⁻¹).subtype ↑1)
b : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := integralBasis K
b' : Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℚ K := (Algebra.traceForm ℚ K).dualBasis ⋯ b
hb : Submodule.span ℤ (Set.range ⇑b) = Submodule.restrictScalars ℤ 1
⊢ AddSubgroup.toIntSubmodule (↑(FractionalIdeal.dual ℤ ℚ 1)).toAddSubgroup =
    (Algebra.traceForm ℚ K).dualSubmodule (Submodule.restrictScalars ℤ 1)
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
⊢ (relNorm R) ⊥ = ⊥

Tactic:
simpa using

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
⊢ (relNorm R) ⊥ = ⊥
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
⊢ (relNorm R) ⊤ = ⊤

Tactic:
simpa using

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
⊢ (relNorm R) ⊤ = ⊤
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 = ⊥
⊢ ∃ s, (relNorm R) P = p ^ s

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 = ⊥
⊢ ∃ s, (relNorm R) P = p ^ s

Tactic:
· refine ⟨1, ?_⟩
    have : P.LiesOver ⊥ := hp ▸ hPp
    rw [hp, eq_bot_of_liesOver_bot R P, relNorm_bot, bot_pow (one_ne_zero)]

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 = ⊥
⊢ ∃ 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 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:
simpa using

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.Prime

Goal state before:
pp: α : Type u_1
inst✝ : CommRing α
p : α
h1 : p ≠ 0
h2 : ¬IsUnit p
h3 : ∀ (a b : α), p ∣ a * b → p ∣ a ∨ p ∣ b
⊢ ∀ (a b : α), -p ∣ a * b → -p ∣ a ∨ -p ∣ b

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝ : CommRing α
p : α
h1 : p ≠ 0
h2 : ¬IsUnit p
h3 : ∀ (a b : α), p ∣ a * b → p ∣ a ∨ p ∣ b
⊢ ∀ (a b : α), p ∣ a * b → p ∣ a ∨ p ∣ b
Module: Mathlib.RingTheory.DedekindDomain.LinearDisjoint

Goal state before:
pp: A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst✝³⁴ : CommRing A
inst✝³³ : Field K
inst✝³² : Algebra A K
inst✝³¹ : IsFractionRing A K
inst✝³⁰ : CommRing B
inst✝²⁹ : Field L
inst✝²⁸ : Algebra B L
inst✝²⁷ : Algebra A L
inst✝²⁶ : Algebra K L
inst✝²⁵ : FiniteDimensional K L
inst✝²⁴ : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst✝²³ : CommRing R₁
inst✝²² : CommRing R₂
inst✝²¹ : IsDomain R₁
inst✝²⁰ : Algebra A R₁
inst✝¹⁹ : Algebra A R₂
inst✝¹⁸ : Algebra R₁ B
inst✝¹⁷ : Algebra R₂ B
inst✝¹⁶ : Algebra R₁ L
inst✝¹⁵ : Algebra R₂ L
inst✝¹⁴ : IsScalarTower A R₁ L
inst✝¹³ : IsScalarTower R₁ B L
inst✝¹² : IsScalarTower R₂ B L
inst✝¹¹ : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst✝¹⁰ : Algebra R₁ ↥F₁
inst✝⁹ : Algebra R₂ ↥F₂
inst✝⁸ : IsTorsionFree R₁ ↥F₁
inst✝⁷ : IsScalarTower A (↥F₂) L
inst✝⁶ : IsScalarTower A R₂ ↥F₂
inst✝⁵ : IsScalarTower R₁ (↥F₁) L
inst✝⁴ : IsScalarTower R₂ (↥F₂) L
inst✝³ : Algebra.IsSeparable K ↥F₂
inst✝² : Algebra.IsSeparable (↥F₁) L
inst✝¹ : Free A R₂
inst✝ : IsLocalization (algebraMapSubmonoid R₂ A⁰) ↥F₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
x : L
hx : x ∈ restrictScalars R₁ (traceDual R₁ (↥F₁) 1)
⊢ F₁.toSubalgebra ⊔ F₂.toSubalgebra = ⊤

Tactic:
simpa using

Goal state after:
A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst✝³⁴ : CommRing A
inst✝³³ : Field K
inst✝³² : Algebra A K
inst✝³¹ : IsFractionRing A K
inst✝³⁰ : CommRing B
inst✝²⁹ : Field L
inst✝²⁸ : Algebra B L
inst✝²⁷ : Algebra A L
inst✝²⁶ : Algebra K L
inst✝²⁵ : FiniteDimensional K L
inst✝²⁴ : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst✝²³ : CommRing R₁
inst✝²² : CommRing R₂
inst✝²¹ : IsDomain R₁
inst✝²⁰ : Algebra A R₁
inst✝¹⁹ : Algebra A R₂
inst✝¹⁸ : Algebra R₁ B
inst✝¹⁷ : Algebra R₂ B
inst✝¹⁶ : Algebra R₁ L
inst✝¹⁵ : Algebra R₂ L
inst✝¹⁴ : IsScalarTower A R₁ L
inst✝¹³ : IsScalarTower R₁ B L
inst✝¹² : IsScalarTower R₂ B L
inst✝¹¹ : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst✝¹⁰ : Algebra R₁ ↥F₁
inst✝⁹ : Algebra R₂ ↥F₂
inst✝⁸ : IsTorsionFree R₁ ↥F₁
inst✝⁷ : IsScalarTower A (↥F₂) L
inst✝⁶ : IsScalarTower A R₂ ↥F₂
inst✝⁵ : IsScalarTower R₁ (↥F₁) L
inst✝⁴ : IsScalarTower R₂ (↥F₂) L
inst✝³ : Algebra.IsSeparable K ↥F₂
inst✝² : Algebra.IsSeparable (↥F₁) L
inst✝¹ : Free A R₂
inst✝ : IsLocalization (algebraMapSubmonoid R₂ A⁰) ↥F₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
x : L
hx : x ∈ restrictScalars R₁ (traceDual R₁ (↥F₁) 1)
⊢ F₁.toSubalgebra ⊔ F₂.toSubalgebra = ⊤
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
x : K
h : IsIntegral ℤ x
u : ℤˣ
n : ℕ
hcycl : IsCyclotomicExtension {p ^ 0} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ 0)
B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ
hint : IsIntegral ℤ B.gen
this : FiniteDimensional ℚ K := finiteDimensional {p ^ 0} ℚ K
hun : Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑p ^ n
H : (algebraMap ℚ K) ↑p ^ n * x ∈ ℤ[B.gen]
⊢ IsCyclotomicExtension {1} ℚ K

Tactic:
simpa using

Goal state after:
p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
x : K
h : IsIntegral ℤ x
u : ℤˣ
n : ℕ
hcycl : IsCyclotomicExtension {p ^ 0} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ 0)
B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ
hint : IsIntegral ℤ B.gen
this : FiniteDimensional ℚ K := finiteDimensional {p ^ 0} ℚ K
hun : Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑p ^ n
H : (algebraMap ℚ K) ↑p ^ n * x ∈ ℤ[B.gen]
⊢ IsCyclotomicExtension {1} ℚ K
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
x : K
h : IsIntegral ℤ x
u : ℤˣ
n n✝ : ℕ
hcycl : IsCyclotomicExtension {p ^ (n✝ + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (n✝ + 1))
B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ
hint : IsIntegral ℤ B.gen
this : FiniteDimensional ℚ K := finiteDimensional {p ^ (n✝ + 1)} ℚ K
hun : Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑p ^ n
H : (algebraMap ℚ K) ↑p ^ n * x ∈ ℤ[B.gen]
hmin : (minpoly ℤ B.gen).IsEisensteinAt (ℤ ∙ ↑p)
⊢ ↑p ^ n • x ∈ ℤ[B.gen]

Tactic:
simpa using

Goal state after:
p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
x : K
h : IsIntegral ℤ x
u : ℤˣ
n n✝ : ℕ
hcycl : IsCyclotomicExtension {p ^ (n✝ + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (n✝ + 1))
B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ
hint : IsIntegral ℤ B.gen
this : FiniteDimensional ℚ K := finiteDimensional {p ^ (n✝ + 1)} ℚ K
hun : Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑p ^ n
H : (algebraMap ℚ K) ↑p ^ n * x ∈ ℤ[B.gen]
hmin : (minpoly ℤ B.gen).IsEisensteinAt (ℤ ∙ ↑p)
⊢ ↑p ^ n * x ∈ ℤ[B.gen]
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: case refine_1
p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
h : hζ.toInteger = 1
⊢ ζ ^ 1 = 1

Tactic:
simpa using

Goal state after:
case refine_1
p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
h : hζ.toInteger = 1
⊢ ζ = 1
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: R : Type u
K : Type v
L : Type z
p : R
inst✝⁹ : CommRing R
inst✝⁸ : Field K
inst✝⁷ : Field L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra R K
inst✝³ : IsScalarTower R K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝ : Module.Finite K L
P : R[X] := minpoly R B.gen
hei : P.IsEisensteinAt (R ∙ p)
hndiv : ¬p ^ 2 ∣ P.coeff 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
this : Module.IsTorsionFree R L
Q₁ : R[X]
hQ : (aeval B.gen).toRingHom Q₁ = p • z
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
⊢ (aeval B.gen) Q = p • z

Tactic:
simpa using

Goal state after:
R : Type u
K : Type v
L : Type z
p : R
inst✝⁹ : CommRing R
inst✝⁸ : Field K
inst✝⁷ : Field L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra R K
inst✝³ : IsScalarTower R K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝ : Module.Finite K L
P : R[X] := minpoly R B.gen
hei : P.IsEisensteinAt (R ∙ p)
hndiv : ¬p ^ 2 ∣ P.coeff 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
this : Module.IsTorsionFree R L
Q₁ : R[X]
hQ : (aeval B.gen).toRingHom Q₁ = p • z
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
⊢ (aeval B.gen) Q₁ = p • z
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
hodd : p ≠ 2
⊢ (Algebra.norm ℤ) (hζ.toInteger - 1) = ↑p

Tactic:
simpa using

Goal state after:
p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
hodd : p ≠ 2
⊢ (Algebra.norm ℤ) (hζ.toInteger - 1) = ↑p
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {2} ℚ K
hζ✝ : IsPrimitiveRoot ζ 2
hζ : IsPrimitiveRoot ζ (2 ^ (0 + 1))
⊢ (Algebra.norm ℤ) (hζ✝.toInteger - 1) = -2

Tactic:
simpa using

Goal state after:
K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {2} ℚ K
hζ✝ : IsPrimitiveRoot ζ 2
hζ : IsPrimitiveRoot ζ (2 ^ (0 + 1))
⊢ (Algebra.norm ℤ) (hζ✝.toInteger - 1) = -2
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p} ℚ K
hζ : IsPrimitiveRoot ζ p
h : p ≠ 2
⊢ IsCyclotomicExtension {p ^ (0 + 1)} ℚ K

Tactic:
simpa using

Goal state after:
p : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p} ℚ K
hζ : IsPrimitiveRoot ζ p
h : p ≠ 2
⊢ IsCyclotomicExtension {p} ℚ K