Theorem: IsCyclotomicExtension.Rat.Three.lambda_dvd_or_dvd_sub_one_or_dvd_add_one
File: Mathlib/NumberTheory/Cyclotomic/Three.lean

tactic: have := hζ.finite_quotient_toInteger_sub_one (by decide)
state_before: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
state_after: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1

tactic: let _ := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
state_before: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
state_after: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝ : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1

tactic: let _ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
state_before: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝ : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
state_after: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1

tactic: let _ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
state_before: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
state_after: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1

tactic: have := Finset.mem_univ (Ideal.Quotient.mk (Ideal.span {λ}) x)
state_before: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
state_after: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this✝ : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
this : (Ideal.Quotient.mk (Ideal.span {λ})) x ∈ Finset.univ
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1

tactic: have h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3 := by
  rw [← Nat.card_eq_fintype_card, hζ.card_quotient_toInteger_sub_one (by decide),
    hζ.norm_toInteger_sub_one_of_prime_ne_two' (by decide)]
  simp only [PNat.val_ofNat, Nat.cast_ofNat, Int.reduceAbs]
state_before: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this✝ : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
this : (Ideal.Quotient.mk (Ideal.span {λ})) x ∈ Finset.univ
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
state_after: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this✝ : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
this : (Ideal.Quotient.mk (Ideal.span {λ})) x ∈ Finset.univ
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1

tactic: rw [Finset.univ_of_card_le_three h3.le] at this
state_before: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this✝ : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
this : (Ideal.Quotient.mk (Ideal.span {λ})) x ∈ Finset.univ
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
state_after: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this✝ : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
this : (Ideal.Quotient.mk (Ideal.span {λ})) x ∈ {0, 1, -1}
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1

tactic: simp only [Finset.mem_insert, Finset.mem_singleton] at this
state_before: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this✝ : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
this : (Ideal.Quotient.mk (Ideal.span {λ})) x ∈ {0, 1, -1}
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
state_after: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this✝ : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
this :
  (Ideal.Quotient.mk (Ideal.span {λ})) x = 0 ∨
    (Ideal.Quotient.mk (Ideal.span {λ})) x = 1 ∨ (Ideal.Quotient.mk (Ideal.span {λ})) x = -1
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1

tactic: rcases this with h | h | h
state_before: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this✝ : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
this :
  (Ideal.Quotient.mk (Ideal.span {λ})) x = 0 ∨
    (Ideal.Quotient.mk (Ideal.span {λ})) x = 1 ∨ (Ideal.Quotient.mk (Ideal.span {λ})) x = -1
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
state_after: case inl
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = 0
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1

case inr.inl
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = 1
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1

case inr.inr
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = -1
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1

tactic: decide
state_before: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
⊢ 1 < 3
state_after: no goals

tactic: rw [← Nat.card_eq_fintype_card, hζ.card_quotient_toInteger_sub_one (by decide),
  hζ.norm_toInteger_sub_one_of_prime_ne_two' (by decide)]
state_before: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this✝ : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
this : (Ideal.Quotient.mk (Ideal.span {λ})) x ∈ Finset.univ
⊢ Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
state_after: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this✝ : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
this : (Ideal.Quotient.mk (Ideal.span {λ})) x ∈ Finset.univ
⊢ (↑↑3).natAbs = 3

tactic: simp only [PNat.val_ofNat, Nat.cast_ofNat, Int.reduceAbs]
state_before: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this✝ : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
this : (Ideal.Quotient.mk (Ideal.span {λ})) x ∈ Finset.univ
⊢ (↑↑3).natAbs = 3
state_after: no goals

tactic: decide
state_before: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this✝ : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
this : (Ideal.Quotient.mk (Ideal.span {λ})) x ∈ Finset.univ
⊢ 1 < 3
state_after: no goals

tactic: decide
state_before: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this✝ : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
this : (Ideal.Quotient.mk (Ideal.span {λ})) x ∈ Finset.univ
⊢ 3 ≠ 2
state_after: no goals

tactic: left
state_before: case inl
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = 0
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
state_after: case inl.h
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = 0
⊢ λ ∣ x

tactic: exact Ideal.mem_span_singleton.1 <| Ideal.Quotient.eq_zero_iff_mem.1 h
state_before: case inl.h
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = 0
⊢ λ ∣ x
state_after: no goals

tactic: right
state_before: case inr.inl
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = 1
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
state_after: case inr.inl.h
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = 1
⊢ λ ∣ x - 1 ∨ λ ∣ x + 1

tactic: left
state_before: case inr.inl.h
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = 1
⊢ λ ∣ x - 1 ∨ λ ∣ x + 1
state_after: case inr.inl.h.h
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = 1
⊢ λ ∣ x - 1

tactic: refine Ideal.mem_span_singleton.1 <| Ideal.Quotient.eq_zero_iff_mem.1 ?_
state_before: case inr.inl.h.h
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = 1
⊢ λ ∣ x - 1
state_after: case inr.inl.h.h
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = 1
⊢ (Ideal.Quotient.mk (Ideal.span {λ})) (x - 1) = 0

tactic: rw [RingHom.map_sub, h, RingHom.map_one, sub_self]
state_before: case inr.inl.h.h
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = 1
⊢ (Ideal.Quotient.mk (Ideal.span {λ})) (x - 1) = 0
state_after: no goals

tactic: right
state_before: case inr.inr
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = -1
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
state_after: case inr.inr.h
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = -1
⊢ λ ∣ x - 1 ∨ λ ∣ x + 1

tactic: right
state_before: case inr.inr.h
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = -1
⊢ λ ∣ x - 1 ∨ λ ∣ x + 1
state_after: case inr.inr.h.h
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = -1
⊢ λ ∣ x + 1

tactic: refine Ideal.mem_span_singleton.1 <| Ideal.Quotient.eq_zero_iff_mem.1 ?_
state_before: case inr.inr.h.h
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = -1
⊢ λ ∣ x + 1
state_after: case inr.inr.h.h
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = -1
⊢ (Ideal.Quotient.mk (Ideal.span {λ})) (x + 1) = 0

tactic: rw [RingHom.map_add, h, RingHom.map_one, add_left_neg]
state_before: case inr.inr.h.h
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
x✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing
x✝ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup
h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3
h : (Ideal.Quotient.mk (Ideal.span {λ})) x = -1
⊢ (Ideal.Quotient.mk (Ideal.span {λ})) (x + 1) = 0
state_after: no goals