awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : Finite.{u_1 + 1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_1 : Fintype (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) :=
  Fintype.ofFinite (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_2 : Ring (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := CommRing.toRing
x_3 : AddGroup (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := AddGroupWithOne.toAddGroup
h3 : Fintype.card.{u_1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) = 3
h :
  Eq (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    (1 : HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
inr :
  HSub.hSub (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) (β :=
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
      ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
        HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
      ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) 1 :
        HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) =
    0
inr₁ :
  HSub.hSub (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) 1
      ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) 1) =
    0
⊢ Eq (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) (x - 1) :
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    (0 : HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
have inr₂ : HSub.hSub (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) 1 1 = 0
Tactic: rw [RingHom.map_sub, h, RingHom.map_one, sub_self]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : Finite.{u_1 + 1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_1 : Fintype (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) :=
  Fintype.ofFinite (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_2 : Ring (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := CommRing.toRing
x_3 : AddGroup (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := AddGroupWithOne.toAddGroup
h3 : Fintype.card.{u_1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) = 3
h :
  Eq (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    (1 : HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
inr :
  HSub.hSub (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) (β :=
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
      ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
        HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
      ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) 1 :
        HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) =
    0
inr₁ :
  HSub.hSub (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) 1
      ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) 1) =
    0
inr₂ : HSub.hSub (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) 1 1 = 0
⊢ Eq (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) (x - 1) :
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    (0 : HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
have inr₃ : Eq.{u_1 + 1} (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) 0 0
Tactic: rw [RingHom.map_sub, h, RingHom.map_one, sub_self]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : Finite.{u_1 + 1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_1 : Fintype (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) :=
  Fintype.ofFinite (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_2 : Ring (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := CommRing.toRing
x_3 : AddGroup (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := AddGroupWithOne.toAddGroup
h3 : Fintype.card.{u_1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) = 3
h :
  Eq (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    (-1 : HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
have inr : λ ∣ x - 1 ∨ λ ∣ x + 1
Tactic: right
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : Finite.{u_1 + 1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_1 : Fintype (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) :=
  Fintype.ofFinite (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_2 : Ring (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := CommRing.toRing
x_3 : AddGroup (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := AddGroupWithOne.toAddGroup
h3 : Fintype.card.{u_1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) = 3
h :
  Eq (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    (-1 : HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
⊢ λ ∣ x - 1 ∨ λ ∣ x + 1
have inr : λ ∣ x + 1
Tactic: right
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : Finite.{u_1 + 1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_1 : Fintype (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) :=
  Fintype.ofFinite (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_2 : Ring (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := CommRing.toRing
x_3 : AddGroup (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := AddGroupWithOne.toAddGroup
h3 : Fintype.card.{u_1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) = 3
h :
  Eq (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    (-1 : HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
⊢ λ ∣ x + 1
have inr : Eq (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
  ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) (x + 1) :
    HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
  (0 : HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
Tactic: refine Ideal.mem_span_singleton.1 <| Ideal.Quotient.eq_zero_iff_mem.1 ?_
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : Finite.{u_1 + 1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_1 : Fintype (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) :=
  Fintype.ofFinite (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_2 : Ring (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := CommRing.toRing
x_3 : AddGroup (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := AddGroupWithOne.toAddGroup
h3 : Fintype.card.{u_1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) = 3
h :
  Eq (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    (-1 : HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
⊢ Eq (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) (x + 1) :
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    (0 : HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
have inr : HAdd.hAdd (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) (β :=
    HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) 1 :
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) =
  0
Tactic: rw [RingHom.map_add, h, RingHom.map_one, neg_add_cancel]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : Finite.{u_1 + 1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_1 : Fintype (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) :=
  Fintype.ofFinite (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_2 : Ring (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := CommRing.toRing
x_3 : AddGroup (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := AddGroupWithOne.toAddGroup
h3 : Fintype.card.{u_1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) = 3
h :
  Eq (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    (-1 : HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
inr :
  HAdd.hAdd (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) (β :=
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
      ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
        HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
      ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) 1 :
        HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) =
    0
⊢ Eq (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) (x + 1) :
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    (0 : HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
have inr₁ : HAdd.hAdd (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) (-1)
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) 1) =
  0
Tactic: rw [RingHom.map_add, h, RingHom.map_one, neg_add_cancel]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : Finite.{u_1 + 1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_1 : Fintype (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) :=
  Fintype.ofFinite (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_2 : Ring (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := CommRing.toRing
x_3 : AddGroup (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := AddGroupWithOne.toAddGroup
h3 : Fintype.card.{u_1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) = 3
h :
  Eq (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    (-1 : HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
inr :
  HAdd.hAdd (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) (β :=
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
      ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
        HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
      ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) 1 :
        HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) =
    0
inr₁ :
  HAdd.hAdd (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) (-1)
      ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) 1) =
    0
⊢ Eq (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) (x + 1) :
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    (0 : HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
have inr₂ : HAdd.hAdd (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) (-1) 1 = 0
Tactic: rw [RingHom.map_add, h, RingHom.map_one, neg_add_cancel]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
this : Finite.{u_1 + 1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_1 : Fintype (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) :=
  Fintype.ofFinite (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
x_2 : Ring (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := CommRing.toRing
x_3 : AddGroup (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) := AddGroupWithOne.toAddGroup
h3 : Fintype.card.{u_1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) = 3
h :
  Eq (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    (-1 : HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
inr :
  HAdd.hAdd (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) (β :=
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
      ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
        HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
      ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) 1 :
        HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) =
    0
inr₁ :
  HAdd.hAdd (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) (-1)
      ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) 1) =
    0
inr₂ : HAdd.hAdd (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) (-1) 1 = 0
⊢ Eq (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) (x + 1) :
      HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
    (0 : HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))
have inr₃ : Eq.{u_1 + 1} (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) 0 0
Tactic: rw [RingHom.map_add, h, RingHom.map_one, neg_add_cancel]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
have have : ∀ (this : Finite.{u_1 + 1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))),
  λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
Tactic: classical
  have := hζ.finite_quotient_toInteger_sub_one (by decide)
  let _ := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
  let _ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing -- to speed up instance synthesis
  let _ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup -- ditto
  have := Finset.mem_univ (Ideal.Quotient.mk (Ideal.span {λ}) x)
  have h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3 := by
    rw [← Nat.card_eq_fintype_card, hζ.card_quotient_toInteger_sub_one,
      hζ.norm_toInteger_sub_one_of_prime_ne_two' (by decide)]
    simp only [Nat.cast_ofNat, Int.reduceAbs]
  rw [Finset.univ_of_card_le_three h3.le] at this
  simp only [Finset.mem_insert, Finset.mem_singleton] at this
  rcases this with h | h | h
  · left
    exact Ideal.mem_span_singleton.1 <| Ideal.Quotient.eq_zero_iff_mem.1 h
  · right; left
    refine Ideal.mem_span_singleton.1 <| Ideal.Quotient.eq_zero_iff_mem.1 ?_
    rw [RingHom.map_sub, h, RingHom.map_one, sub_self]
  · right; right
    refine Ideal.mem_span_singleton.1 <| Ideal.Quotient.eq_zero_iff_mem.1 ?_
    rw [RingHom.map_add, h, RingHom.map_one, neg_add_cancel]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
have have₁ have₂ have₃ have₄ have₅ :
  ∀ (this : Finite.{u_1 + 1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))),
    λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
have have₆ : ∀ (this : Finite.{u_1 + 1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))),
  let x_1 : Fintype (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) :=
    Fintype.ofFinite (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}));
  ∀
    (this :
      Membership.mem (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) (γ :=
        Finset (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))) Finset.univ
        ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
          HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))),
    λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
Tactic: classical
  have := hζ.finite_quotient_toInteger_sub_one (by decide)
  let _ := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
  let _ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing -- to speed up instance synthesis
  let _ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup -- ditto
  have := Finset.mem_univ (Ideal.Quotient.mk (Ideal.span {λ}) x)
  have h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3 := by
    rw [← Nat.card_eq_fintype_card, hζ.card_quotient_toInteger_sub_one,
      hζ.norm_toInteger_sub_one_of_prime_ne_two' (by decide)]
    simp only [Nat.cast_ofNat, Int.reduceAbs]
  rw [Finset.univ_of_card_le_three h3.le] at this
  simp only [Finset.mem_insert, Finset.mem_singleton] at this
  rcases this with h | h | h
  · left
    exact Ideal.mem_span_singleton.1 <| Ideal.Quotient.eq_zero_iff_mem.1 h
  · right; left
    refine Ideal.mem_span_singleton.1 <| Ideal.Quotient.eq_zero_iff_mem.1 ?_
    rw [RingHom.map_sub, h, RingHom.map_one, sub_self]
  · right; right
    refine Ideal.mem_span_singleton.1 <| Ideal.Quotient.eq_zero_iff_mem.1 ?_
    rw [RingHom.map_add, h, RingHom.map_one, neg_add_cancel]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
x : 𝓞 K
inst_1 : NumberField K
inst_2 : IsCyclotomicExtension {3} ℚ K
have have₁ have₂ have₃ have₄ have₅ :
  ∀ (this : Finite.{u_1 + 1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))),
    λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
have₆ :
  ∀ (this : Finite.{u_1 + 1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))),
    let x_1 : Fintype (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) :=
      Fintype.ofFinite (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}));
    ∀
      (this :
        Membership.mem (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) (γ :=
          Finset (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))) Finset.univ
          ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
            HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))),
      λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
have have₇ : ∀ (this : Finite.{u_1 + 1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))),
  let x_1 : Fintype (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) :=
    Fintype.ofFinite (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}));
  ∀
    (this :
      Membership.mem (α := HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) (γ :=
        Finset (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ}))) Finset.univ
        ((Ideal.Quotient.mk (Ideal.span {λ}) : (a : 𝓞 K) → 𝓞 K ⧸ Ideal.span {λ}) x :
          HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})))
    (h3 : Fintype.card.{u_1} (HasQuotient.Quotient (𝓞 K) (B := Ideal (𝓞 K)) (Ideal.span {λ})) = 3),
    λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1
Tactic: classical
  have := hζ.finite_quotient_toInteger_sub_one (by decide)
  let _ := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
  let _ : Ring (𝓞 K ⧸ Ideal.span {λ}) := CommRing.toRing -- to speed up instance synthesis
  let _ : AddGroup (𝓞 K ⧸ Ideal.span {λ}) := AddGroupWithOne.toAddGroup -- ditto
  have := Finset.mem_univ (Ideal.Quotient.mk (Ideal.span {λ}) x)
  have h3 : Fintype.card (𝓞 K ⧸ Ideal.span {λ}) = 3 := by
    rw [← Nat.card_eq_fintype_card, hζ.card_quotient_toInteger_sub_one,
      hζ.norm_toInteger_sub_one_of_prime_ne_two' (by decide)]
    simp only [Nat.cast_ofNat, Int.reduceAbs]
  rw [Finset.univ_of_card_le_three h3.le] at this
  simp only [Finset.mem_insert, Finset.mem_singleton] at this
  rcases this with h | h | h
  · left
    exact Ideal.mem_span_singleton.1 <| Ideal.Quotient.eq_zero_iff_mem.1 h
  · right; left
    refine Ideal.mem_span_singleton.1 <| Ideal.Quotient.eq_zero_iff_mem.1 ?_
    rw [RingHom.map_sub, h, RingHom.map_one, sub_self]
  · right; right
    refine Ideal.mem_span_singleton.1 <| Ideal.Quotient.eq_zero_iff_mem.1 ?_
    rw [RingHom.map_add, h, RingHom.map_one, neg_add_cancel]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Cyclotomic.Three (goal)
Goal:
K : Type u_1
inst : Field K
ζ : K
hζ : IsPrimitiveRoot ζ 3
⊢ HPow.hPow (β := ℕ) (↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) 2 + (↑(IsUnit.unit (a := hζ.toInteger) ⋯) : 𝓞 K) + 1 =
    0
have rw : -(↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) - 1 + (↑(IsUnit.unit (a := hζ.toInteger) sorry) : 𝓞 K) + 1 = 0
Tactic: rw [eta_sq]