internlm/Lean-Github huggingface dataset:
Theorem: Hilbert92ish
Proof state before:
case NE_p_pow.a.h
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H
inst✝⁷ : Fintype H
hCard : Fintype.card H = ↑p
σ✝ : H
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
h : ℕ
ζ : (𝓞 k)ˣ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ h)
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
⊢ ↑↑((Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) NE) = ↑↑(E ^ ↑p)
Tactic:
simp only [RingHom.toMonoidHom_eq_coe, Units.coe_map, MonoidHom.coe_coe,
  RingOfInteger.coe_algebraMap_apply, Units.val_pow_eq_pow_val, map_pow]
Proof state after:
case NE_p_pow.a.h
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H
inst✝⁷ : Fintype H
hCard : Fintype.card H = ↑p
σ✝ : H
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
h : ℕ
ζ : (𝓞 k)ˣ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ h)
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
⊢ (algebraMap k K) ↑↑NE = (algebraMap (𝓞 K) K) ↑E ^ ↑p
Theorem: Hilbert92ish
Proof state before:
case NE_p_pow.a.h
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H
inst✝⁷ : Fintype H
hCard : Fintype.card H = ↑p
σ✝ : H
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
h : ℕ
ζ : (𝓞 k)ˣ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ h)
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
⊢ (algebraMap k K) ↑↑NE = (algebraMap (𝓞 K) K) ↑E ^ ↑p
Tactic:
rw [← map_pow] at hE
Proof state after:
case NE_p_pow.a.h
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H
inst✝⁷ : Fintype H
hCard : Fintype.card H = ↑p
σ✝ : H
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
h : ℕ
ζ : (𝓞 k)ˣ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ h)
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ (h - 1)) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
⊢ (algebraMap k K) ↑↑NE = (algebraMap (𝓞 K) K) ↑E ^ ↑p
Theorem: Hilbert92ish
Proof state before:
case NE_p_pow.a.h
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H
inst✝⁷ : Fintype H
hCard : Fintype.card H = ↑p
σ✝ : H
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
h : ℕ
ζ : (𝓞 k)ˣ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ h)
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ (h - 1)) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
⊢ (algebraMap k K) ↑↑NE = (algebraMap (𝓞 K) K) ↑E ^ ↑p
Tactic:
refine Hilbert92ish_aux2 p hp hKL σ hσ E _ hE ?_ hpodd
Proof state after:
case NE_p_pow.a.h
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H
inst✝⁷ : Fintype H
hCard : Fintype.card H = ↑p
σ✝ : H
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
h : ℕ
ζ : (𝓞 k)ˣ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ h)
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ (h - 1)) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
⊢ ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ (h - 1)) ^ ↑p = 1
Theorem: Hilbert92ish
Proof state before:
case NE_p_pow.a.h
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H
inst✝⁷ : Fintype H
hCard : Fintype.card H = ↑p
σ✝ : H
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
h : ℕ
ζ : (𝓞 k)ˣ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ h)
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ (h - 1)) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
⊢ ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ (h - 1)) ^ ↑p = 1
Tactic:
rw [← pow_mul, ← pow_succ]
Proof state after:
case NE_p_pow.a.h
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H
inst✝⁷ : Fintype H
hCard : Fintype.card H = ↑p
σ✝ : H
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
h : ℕ
ζ : (𝓞 k)ˣ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ h)
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ (h - 1)) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
⊢ (algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ (h - 1 + 1) = 1
Theorem: Hilbert92ish
Proof state before:
case NE_p_pow.a.h
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H
inst✝⁷ : Fintype H
hCard : Fintype.card H = ↑p
σ✝ : H
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
h : ℕ
ζ : (𝓞 k)ˣ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ h)
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ (h - 1)) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
⊢ (algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ (h - 1 + 1) = 1
Tactic:
apply (hζ.pow_eq_one_iff_dvd _).2
Proof state after:
case NE_p_pow.a.h
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H
inst✝⁷ : Fintype H
hCard : Fintype.card H = ↑p
σ✝ : H
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
h : ℕ
ζ : (𝓞 k)ˣ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ h)
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ (h - 1)) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
⊢ ↑p ^ h ∣ ↑p ^ (h - 1 + 1)
Theorem: Hilbert92ish
Proof state before:
case NE_p_pow.a.h
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H
inst✝⁷ : Fintype H
hCard : Fintype.card H = ↑p
σ✝ : H
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
h : ℕ
ζ : (𝓞 k)ˣ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ h)
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ ^ ↑p ^ (h - 1)) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
⊢ ↑p ^ h ∣ ↑p ^ (h - 1 + 1)
Tactic:
cases h <;> simp only [Nat.zero_eq, pow_zero, zero_le, tsub_eq_zero_of_le,
  zero_add, pow_one, one_dvd, Nat.succ_sub_succ_eq_sub,
  nonpos_iff_eq_zero, tsub_zero, dvd_refl]
Proof state after:
no goals
Theorem: Hilbert92ish
Proof state before:
case neg.intro.intro.intro.intro.intro.intro.intro.left
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H✝ : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H✝
inst✝⁷ : Fintype H✝
hCard : Fintype.card H✝ = ↑p
σ✝ : H✝
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H✝ G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
h : ℕ
ζ : (𝓞 k)ˣ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ h)
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
NE_p_pow : (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) NE = E ^ ↑p
H : Fin (r + 1) → Additive (𝓞 K)ˣ := unitlifts p hp hKL σ hσ S
N : Fin (r + 1) → Additive (𝓞 k)ˣ :=
  fun e => Additive.ofMul ((Units.map (RingOfIntegers.norm k)) (Additive.toMul (H e)))
η : Fin (r + 2) → Additive (𝓞 k)ˣ := Fin.snoc N (Additive.ofMul NE)
a : ℤ
ι : Fin (r + 2) → ℤ
i✝ : Fin (r + 2)
ha : ∑ i : Fin (r + 2), ι i • η i = (a * ↑↑p) • Additive.ofMul ζ
ha' : ¬↑↑p ∣ ι i✝
ha'' : ζ = 1 → i✝ ≠ Fin.last (r + 1)
H2 : Fin (r + 2) → Additive (𝓞 K)ˣ := Fin.snoc H (Additive.ofMul E)
J : (𝓞 K)ˣ := Additive.toMul (∑ i : Fin (r + 2), ι i • H2 i) * (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ζ ^ (-a)
i : Fin (r + 2)
⊢ (algebraMap (𝓞 k) k) ↑(Additive.toMul (η i)) = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑(Additive.toMul (H2 i)))
Tactic:
induction i using Fin.lastCases with
| last =>
  simp only [Fin.snoc_last, toMul_ofMul, Units.coe_map, RingOfIntegers.coe_norm, NE, η, H2]
| cast i =>
  simp only [Fin.snoc_castSucc, toMul_ofMul, Units.coe_map, RingOfIntegers.coe_norm, NE,
    η, H2, J, N, H]
Proof state after:
no goals
Theorem: Hilbert92ish
Proof state before:
case neg.intro.intro.intro.intro.intro.intro.intro.left.last
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H✝ : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H✝
inst✝⁷ : Fintype H✝
hCard : Fintype.card H✝ = ↑p
σ✝ : H✝
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H✝ G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
h : ℕ
ζ : (𝓞 k)ˣ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ h)
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
NE_p_pow : (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) NE = E ^ ↑p
H : Fin (r + 1) → Additive (𝓞 K)ˣ := unitlifts p hp hKL σ hσ S
N : Fin (r + 1) → Additive (𝓞 k)ˣ :=
  fun e => Additive.ofMul ((Units.map (RingOfIntegers.norm k)) (Additive.toMul (H e)))
η : Fin (r + 2) → Additive (𝓞 k)ˣ := Fin.snoc N (Additive.ofMul NE)
a : ℤ
ι : Fin (r + 2) → ℤ
i : Fin (r + 2)
ha : ∑ i : Fin (r + 2), ι i • η i = (a * ↑↑p) • Additive.ofMul ζ
ha' : ¬↑↑p ∣ ι i
ha'' : ζ = 1 → i ≠ Fin.last (r + 1)
H2 : Fin (r + 2) → Additive (𝓞 K)ˣ := Fin.snoc H (Additive.ofMul E)
J : (𝓞 K)ˣ := Additive.toMul (∑ i : Fin (r + 2), ι i • H2 i) * (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ζ ^ (-a)
⊢ (algebraMap (𝓞 k) k) ↑(Additive.toMul (η (Fin.last (r + 1)))) =
    (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑(Additive.toMul (H2 (Fin.last (r + 1)))))
Tactic:
simp only [Fin.snoc_last, toMul_ofMul, Units.coe_map, RingOfIntegers.coe_norm, NE, η, H2]
Proof state after:
no goals
Theorem: Hilbert92ish
Proof state before:
case neg.intro.intro.intro.intro.intro.intro.intro.left.cast
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H✝ : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H✝
inst✝⁷ : Fintype H✝
hCard : Fintype.card H✝ = ↑p
σ✝ : H✝
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H✝ G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
hpodd : ↑p ≠ 2
h : ℕ
ζ : (𝓞 k)ˣ
hζ : IsPrimitiveRoot ((algebraMap (𝓞 k) k) ↑ζ) (↑p ^ h)
hζ' : ∀ (ε : (𝓞 k)ˣ) (n : ℕ), ε ^ ↑p ^ n = 1 → ∃ i, ζ ^ i = ε
E : (𝓞 K)ˣ
hE : (algebraMap k K) ((algebraMap (𝓞 k) k) ↑ζ) ^ ↑p ^ (h - 1) = (algebraMap (𝓞 K) K) ↑E / σ ((algebraMap (𝓞 K) K) ↑E)
NE : (𝓞 k)ˣ := (Units.map (RingOfIntegers.norm k)) E
hNE : (algebraMap (𝓞 k) k) ↑NE = (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑E)
S :
  systemOfUnits p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1)
hS :
  systemOfUnits.IsFundamental p
    (Additive (relativeUnitsWithGenerator p hp hKL σ hσ) ⧸
      AddCommGroup.torsion (Additive (relativeUnitsWithGenerator p hp hKL σ hσ)))
    (r + 1) S
NE_p_pow : (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) NE = E ^ ↑p
H : Fin (r + 1) → Additive (𝓞 K)ˣ := unitlifts p hp hKL σ hσ S
N : Fin (r + 1) → Additive (𝓞 k)ˣ :=
  fun e => Additive.ofMul ((Units.map (RingOfIntegers.norm k)) (Additive.toMul (H e)))
η : Fin (r + 2) → Additive (𝓞 k)ˣ := Fin.snoc N (Additive.ofMul NE)
a : ℤ
ι : Fin (r + 2) → ℤ
i✝ : Fin (r + 2)
ha : ∑ i : Fin (r + 2), ι i • η i = (a * ↑↑p) • Additive.ofMul ζ
ha' : ¬↑↑p ∣ ι i✝
ha'' : ζ = 1 → i✝ ≠ Fin.last (r + 1)
H2 : Fin (r + 2) → Additive (𝓞 K)ˣ := Fin.snoc H (Additive.ofMul E)
J : (𝓞 K)ˣ := Additive.toMul (∑ i : Fin (r + 2), ι i • H2 i) * (Units.map ↑(algebraMap (𝓞 k) (𝓞 K))) ζ ^ (-a)
i : Fin (r + 1)
⊢ (algebraMap (𝓞 k) k) ↑(Additive.toMul (η i.castSucc)) =
    (Algebra.norm k) ((algebraMap (𝓞 K) K) ↑(Additive.toMul (H2 i.castSucc)))
Tactic:
simp only [Fin.snoc_castSucc, toMul_ofMul, Units.coe_map, RingOfIntegers.coe_norm, NE,
  η, H2, J, N, H]
Proof state after:
no goals