awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
w : InfinitePlace K
hw : w.IsComplex
x x_1 :
  { x : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } //
    mkComplex x = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw }
h :
  Eq (α := { x : K →+* ℂ // mk x = w })
    (match (motive :=
        { x : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } //
            mkComplex x = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw } →
          { x : K →+* ℂ // mk x = w })
        x with
      | ⟨φ, hφ⟩ => Subtype.mk (p := fun (x : K →+* ℂ) ↦ mk x = w) (↑φ : K →+* ℂ) ⋯ :
      { x : K →+* ℂ // mk x = w })
    (match (motive :=
        { x : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } //
            mkComplex x = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw } →
          { x : K →+* ℂ // mk x = w })
        x_1 with
      | ⟨φ, hφ⟩ => Subtype.mk (p := fun (x : K →+* ℂ) ↦ mk x = w) (↑φ : K →+* ℂ) ⋯ :
      { x : K →+* ℂ // mk x = w })
h₁ :
  ∀
    (h :
      (↑(↑x : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ }) : K →+* ℂ) =
        (↑(↑x_1 : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ }) : K →+* ℂ)),
    x = x_1
⊢ x = x_1
have h₂ : ∀ (h : (↑x : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ }) = (↑x_1 : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })),
  x = x_1
Tactic: rwa [Subtype.mk_eq_mk, ← Subtype.ext_iff, ← Subtype.ext_iff] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
w : InfinitePlace K
hw : w.IsComplex
x x_1 :
  { x : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } //
    mkComplex x = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw }
h :
  Eq (α := { x : K →+* ℂ // mk x = w })
    (match (motive :=
        { x : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } //
            mkComplex x = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw } →
          { x : K →+* ℂ // mk x = w })
        x with
      | ⟨φ, hφ⟩ => Subtype.mk (p := fun (x : K →+* ℂ) ↦ mk x = w) (↑φ : K →+* ℂ) ⋯ :
      { x : K →+* ℂ // mk x = w })
    (match (motive :=
        { x : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } //
            mkComplex x = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw } →
          { x : K →+* ℂ // mk x = w })
        x_1 with
      | ⟨φ, hφ⟩ => Subtype.mk (p := fun (x : K →+* ℂ) ↦ mk x = w) (↑φ : K →+* ℂ) ⋯ :
      { x : K →+* ℂ // mk x = w })
h₁ :
  ∀
    (h :
      (↑(↑x : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ }) : K →+* ℂ) =
        (↑(↑x_1 : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ }) : K →+* ℂ)),
    x = x_1
h₂ :
  ∀ (h : (↑x : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ }) = (↑x_1 : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })),
    x = x_1
⊢ x = x_1
have h₃ : ∀ (h : x = x_1), x = x_1
Tactic: rwa [Subtype.mk_eq_mk, ← Subtype.ext_iff, ← Subtype.ext_iff] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
w : InfinitePlace K
hw : w.IsComplex
x : { x : K →+* ℂ // mk x = w }
φ : K →+* ℂ
hφ : mk φ = w
⊢ ∃ (a :
    { x : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } //
      mkComplex x = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw }),
    Eq (α := { x : K →+* ℂ // mk x = w })
      (match (motive :=
          { x : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } //
              mkComplex x = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw } →
            { x : K →+* ℂ // mk x = w })
          a with
        | ⟨φ, hφ⟩ => Subtype.mk (p := fun (x : K →+* ℂ) ↦ mk x = w) (↑φ : K →+* ℂ) ⋯ :
        { x : K →+* ℂ // mk x = w })
      (Subtype.mk (p := fun (x : K →+* ℂ) ↦ mk x = w) φ hφ : { x : K →+* ℂ // mk x = w })
have h : mkComplex (Subtype.mk (p := fun (φ : K →+* ℂ) ↦ ¬ComplexEmbedding.IsReal φ) φ sorry) =
  Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw
Tactic: refine ⟨⟨⟨φ, not_isReal_of_mk_isComplex (hφ.symm ▸ hw)⟩, ?_⟩, rfl⟩
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
w : InfinitePlace K
hw : w.IsComplex
x : { x : K →+* ℂ // mk x = w }
φ : K →+* ℂ
hφ : mk φ = w
⊢ mkComplex (Subtype.mk (p := fun (φ : K →+* ℂ) ↦ ¬ComplexEmbedding.IsReal φ) φ ⋯) =
    Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw
have h : (↑(mkComplex (Subtype.mk (p := fun (φ : K →+* ℂ) ↦ ¬ComplexEmbedding.IsReal φ) φ sorry)) : InfinitePlace K) =
  (↑(Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw) : InfinitePlace K)
Tactic: rwa [Subtype.ext_iff, mkComplex_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
w : InfinitePlace K
hw : w.IsComplex
x : { x : K →+* ℂ // mk x = w }
φ : K →+* ℂ
hφ : mk φ = w
h :
  (↑(mkComplex (Subtype.mk (p := fun (φ : K →+* ℂ) ↦ ¬ComplexEmbedding.IsReal φ) φ ⋯)) : InfinitePlace K) =
    (↑(Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw) : InfinitePlace K)
⊢ mkComplex (Subtype.mk (p := fun (φ : K →+* ℂ) ↦ ¬ComplexEmbedding.IsReal φ) φ ⋯) =
    Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw
have h₁ : mk (↑(Subtype.mk (p := fun (φ : K →+* ℂ) ↦ ¬ComplexEmbedding.IsReal φ) φ sorry) : K →+* ℂ) =
  (↑(Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw) : InfinitePlace K)
Tactic: rwa [Subtype.ext_iff, mkComplex_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
w : InfinitePlace K
hw : w.IsComplex
⊢ 2 = w.mult
have h : Eq (α := ℕ) 2 (if w.IsReal then 1 else 2)
Tactic: simp_rw [mult, not_isReal_iff_isComplex.mpr hw, ite_false]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
w : InfinitePlace K
hw : w.IsComplex
h : Eq (α := ℕ) 2 (if w.IsReal then 1 else 2)
⊢ 2 = w.mult
have h₁ : Eq (α := ℕ) 2 (if False then 1 else 2)
Tactic: simp_rw [mult, not_isReal_iff_isComplex.mpr hw, ite_false]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
⊢ nrRealPlaces K + 2 * nrComplexPlaces K = finrank ℚ K
have rw : card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } ≤ card.{u_1} (K →+* ℂ)
Tactic: rw [← card_real_embeddings, ← card_complex_embeddings, Fintype.card_subtype_compl,
    ← Embeddings.card K ℂ, Nat.add_sub_of_le]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
⊢ nrRealPlaces K + 2 * nrComplexPlaces K = finrank ℚ K
have rw : card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } + 2 * nrComplexPlaces K = finrank ℚ K
Tactic: classical
  rw [← card_real_embeddings, ← card_complex_embeddings, Fintype.card_subtype_compl,
    ← Embeddings.card K ℂ, Nat.add_sub_of_le]
  exact Fintype.card_subtype_le _
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
rw : card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } + 2 * nrComplexPlaces K = finrank ℚ K
⊢ nrRealPlaces K + 2 * nrComplexPlaces K = finrank ℚ K
have rw₁ : card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } + card.{u_1} { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } =
  finrank ℚ K
Tactic: classical
  rw [← card_real_embeddings, ← card_complex_embeddings, Fintype.card_subtype_compl,
    ← Embeddings.card K ℂ, Nat.add_sub_of_le]
  exact Fintype.card_subtype_le _
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
rw : card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } + 2 * nrComplexPlaces K = finrank ℚ K
rw₁ :
  card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } + card.{u_1} { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } =
    finrank ℚ K
⊢ nrRealPlaces K + 2 * nrComplexPlaces K = finrank ℚ K
have rw₂ : card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } +
    (card.{u_1} (K →+* ℂ) - card.{u_1} { x : K →+* ℂ // ComplexEmbedding.IsReal x }) =
  finrank ℚ K
Tactic: classical
  rw [← card_real_embeddings, ← card_complex_embeddings, Fintype.card_subtype_compl,
    ← Embeddings.card K ℂ, Nat.add_sub_of_le]
  exact Fintype.card_subtype_le _
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
rw : card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } + 2 * nrComplexPlaces K = finrank ℚ K
rw₁ :
  card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } + card.{u_1} { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } =
    finrank ℚ K
rw₂ :
  card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } +
      (card.{u_1} (K →+* ℂ) - card.{u_1} { x : K →+* ℂ // ComplexEmbedding.IsReal x }) =
    finrank ℚ K
⊢ nrRealPlaces K + 2 * nrComplexPlaces K = finrank ℚ K
have rw₃ : card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } +
    (card.{u_1} (K →+* ℂ) - card.{u_1} { x : K →+* ℂ // ComplexEmbedding.IsReal x }) =
  card.{u_1} (K →+* ℂ)
Tactic: classical
  rw [← card_real_embeddings, ← card_complex_embeddings, Fintype.card_subtype_compl,
    ← Embeddings.card K ℂ, Nat.add_sub_of_le]
  exact Fintype.card_subtype_le _
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
rw : card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } + 2 * nrComplexPlaces K = finrank ℚ K
rw₁ :
  card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } + card.{u_1} { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } =
    finrank ℚ K
rw₂ :
  card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } +
      (card.{u_1} (K →+* ℂ) - card.{u_1} { x : K →+* ℂ // ComplexEmbedding.IsReal x }) =
    finrank ℚ K
rw₃ :
  card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } +
      (card.{u_1} (K →+* ℂ) - card.{u_1} { x : K →+* ℂ // ComplexEmbedding.IsReal x }) =
    card.{u_1} (K →+* ℂ)
⊢ nrRealPlaces K + 2 * nrComplexPlaces K = finrank ℚ K
have rw₄ : card.{u_1} (K →+* ℂ) = card.{u_1} (K →+* ℂ)
Tactic: classical
  rw [← card_real_embeddings, ← card_complex_embeddings, Fintype.card_subtype_compl,
    ← Embeddings.card K ℂ, Nat.add_sub_of_le]
  exact Fintype.card_subtype_le _
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
rw : card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } + 2 * nrComplexPlaces K = finrank ℚ K
rw₁ :
  card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } + card.{u_1} { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } =
    finrank ℚ K
rw₂ :
  card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } +
      (card.{u_1} (K →+* ℂ) - card.{u_1} { x : K →+* ℂ // ComplexEmbedding.IsReal x }) =
    finrank ℚ K
rw₃ :
  card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } +
      (card.{u_1} (K →+* ℂ) - card.{u_1} { x : K →+* ℂ // ComplexEmbedding.IsReal x }) =
    card.{u_1} (K →+* ℂ)
rw₄ : card.{u_1} (K →+* ℂ) = card.{u_1} (K →+* ℂ)
⊢ nrRealPlaces K + 2 * nrComplexPlaces K = finrank ℚ K
have rw₅ : card.{u_1} { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } ≤ card.{u_1} (K →+* ℂ)
Tactic: classical
  rw [← card_real_embeddings, ← card_complex_embeddings, Fintype.card_subtype_compl,
    ← Embeddings.card K ℂ, Nat.add_sub_of_le]
  exact Fintype.card_subtype_le _
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
⊢ ((Equiv.Perm.sign : Equiv.Perm (K →+* ℂ) →* ℤˣ) : (a : Equiv.Perm (K →+* ℂ)) → ℤˣ)
      (Function.Involutive.toPerm ComplexEmbedding.conjugate ⋯) =
    HPow.hPow (α := ℤˣ) (-1) (nrComplexPlaces K)
have rw : HPow.hPow (α := Equiv.Perm (K →+* ℂ)) (β := ℕ) (Function.Involutive.toPerm ComplexEmbedding.conjugate sorry) 2 = 1
Tactic: rw [Equiv.Perm.sign_of_pow_two_eq_one, Embeddings.card, ← card_add_two_mul_card_eq_rank,
    ← card_real_embeddings, Fintype.card, Fintype.card, Nat.add_sub_cancel_left,
    Nat.mul_div_cancel_left _ zero_lt_two]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
h : finrank ℚ K = 1
⊢ nrComplexPlaces K = 0
have linarith : nrComplexPlaces K < 0 →
  -1 + HSub.hSub (α := ℤ) 0 (↑(nrComplexPlaces K) : ℤ) + (HAdd.hAdd (α := ℤ) (↑(nrComplexPlaces K) : ℤ) 1 - 0) = 0
Tactic: linarith [card_add_two_mul_card_eq_rank K]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
h : finrank ℚ K = 1
linarith :
  nrComplexPlaces K < 0 →
    -1 + HSub.hSub (α := ℤ) 0 (↑(nrComplexPlaces K) : ℤ) + (HAdd.hAdd (α := ℤ) (↑(nrComplexPlaces K) : ℤ) 1 - 0) = 0
⊢ nrComplexPlaces K = 0
have linarith₁ : 0 < nrComplexPlaces K →
  -1 + HSub.hSub (α := ℤ) 0 (↑(nrRealPlaces K) : ℤ) + HSub.hSub (α := ℤ) (↑(finrank ℚ K) : ℤ) 1 +
        2 * (HAdd.hAdd (α := ℤ) 0 1 - (↑(nrComplexPlaces K) : ℤ)) +
      ((↑(nrRealPlaces K) : ℤ) + HMul.hMul (α := ℤ) 2 (↑(nrComplexPlaces K) : ℤ) - (↑(finrank ℚ K) : ℤ)) =
    0
Tactic: linarith [card_add_two_mul_card_eq_rank K]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
h : finrank ℚ K = 1
linarith :
  nrComplexPlaces K < 0 →
    -1 + HSub.hSub (α := ℤ) 0 (↑(nrComplexPlaces K) : ℤ) + (HAdd.hAdd (α := ℤ) (↑(nrComplexPlaces K) : ℤ) 1 - 0) = 0
linarith₁ :
  0 < nrComplexPlaces K →
    -1 + HSub.hSub (α := ℤ) 0 (↑(nrRealPlaces K) : ℤ) + HSub.hSub (α := ℤ) (↑(finrank ℚ K) : ℤ) 1 +
          2 * (HAdd.hAdd (α := ℤ) 0 1 - (↑(nrComplexPlaces K) : ℤ)) +
        ((↑(nrRealPlaces K) : ℤ) + HMul.hMul (α := ℤ) 2 (↑(nrComplexPlaces K) : ℤ) - (↑(finrank ℚ K) : ℤ)) =
      0
⊢ nrComplexPlaces K = 0
have linarith₂ : 0 < nrComplexPlaces K → GT.gt (α := ℤ) 2 0
Tactic: linarith [card_add_two_mul_card_eq_rank K]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
h : finrank ℚ K = 1
⊢ nrRealPlaces K = 1
have this : nrRealPlaces K + 2 * nrComplexPlaces K = finrank ℚ K
Tactic: have := card_add_two_mul_card_eq_rank K
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
h : finrank ℚ K = 1
this : nrRealPlaces K + 2 * nrComplexPlaces K = finrank ℚ K
⊢ nrRealPlaces K = 1
have rwa : ∀ (this : nrRealPlaces K + HMul.hMul (α := ℕ) 2 0 = finrank ℚ K), nrRealPlaces K = 1
Tactic: rwa [nrComplexPlaces_eq_zero_of_finrank_eq_one h, h, mul_zero, add_zero] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
h : finrank ℚ K = 1
this : nrRealPlaces K + 2 * nrComplexPlaces K = finrank ℚ K
rwa : ∀ (this : nrRealPlaces K + HMul.hMul (α := ℕ) 2 0 = finrank ℚ K), nrRealPlaces K = 1
⊢ nrRealPlaces K = 1
have rwa₁ : ∀ (this : nrRealPlaces K + HMul.hMul (α := ℕ) 2 0 = 1), nrRealPlaces K = 1
Tactic: rwa [nrComplexPlaces_eq_zero_of_finrank_eq_one h, h, mul_zero, add_zero] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
h : finrank ℚ K = 1
this : nrRealPlaces K + 2 * nrComplexPlaces K = finrank ℚ K
rwa : ∀ (this : nrRealPlaces K + HMul.hMul (α := ℕ) 2 0 = finrank ℚ K), nrRealPlaces K = 1
rwa₁ : ∀ (this : nrRealPlaces K + HMul.hMul (α := ℕ) 2 0 = 1), nrRealPlaces K = 1
⊢ nrRealPlaces K = 1
have rwa₂ : ∀ (this : nrRealPlaces K + 0 = 1), nrRealPlaces K = 1
Tactic: rwa [nrComplexPlaces_eq_zero_of_finrank_eq_one h, h, mul_zero, add_zero] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
h : finrank ℚ K = 1
this : nrRealPlaces K + 2 * nrComplexPlaces K = finrank ℚ K
rwa : ∀ (this : nrRealPlaces K + HMul.hMul (α := ℕ) 2 0 = finrank ℚ K), nrRealPlaces K = 1
rwa₁ : ∀ (this : nrRealPlaces K + HMul.hMul (α := ℕ) 2 0 = 1), nrRealPlaces K = 1
rwa₂ : ∀ (this : nrRealPlaces K + 0 = 1), nrRealPlaces K = 1
⊢ nrRealPlaces K = 1
have rwa₃ : ∀ (this : nrRealPlaces K = 1), nrRealPlaces K = 1
Tactic: rwa [nrComplexPlaces_eq_zero_of_finrank_eq_one h, h, mul_zero, add_zero] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
h : Odd (finrank ℚ K)
⊢ 0 < nrRealPlaces K
have refine : nrRealPlaces K ≠ 0
Tactic: refine Nat.pos_of_ne_zero ?_
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
h : Odd (finrank ℚ K)
hc : nrRealPlaces K = 0
⊢ False
have refine : Even (finrank ℚ K)
Tactic: refine (Nat.not_odd_iff_even.mpr ?_) h
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
h : Odd (finrank ℚ K)
hc : nrRealPlaces K = 0
⊢ Even (finrank ℚ K)
have rw : Even (2 * nrComplexPlaces K)
Tactic: rw [← card_add_two_mul_card_eq_rank, hc, zero_add]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
⊢ nrRealPlaces K = 0
have refine : ∀ (x : { w : InfinitePlace K // w.IsReal }) (w : InfinitePlace K) (hwreal : w.IsReal), False
Tactic: refine (@Fintype.card_eq_zero_iff _ (_)).2 ⟨fun ⟨w, hwreal⟩ ↦ ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
x : { w : InfinitePlace K // w.IsReal }
w : InfinitePlace K
hwreal : w.IsReal
⊢ False
have hwreal₁ : ComplexEmbedding.IsReal w.embedding
Tactic: rw [NumberField.InfinitePlace.isReal_iff] at hwreal
Cleared hypotheses: hwreal
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
x : { w : InfinitePlace K // w.IsReal }
w : InfinitePlace K
hwreal : ComplexEmbedding.IsReal w.embedding
f : K →+* ℂ := w.embedding
⊢ False
have hζ' : IsPrimitiveRoot (M := ℂ) ((f : (a : K) → ℂ) ζ : ℂ) k
Tactic: have hζ' : IsPrimitiveRoot (f ζ) k := hζ.map_of_injective f.injective
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
x : { w : InfinitePlace K // w.IsReal }
w : InfinitePlace K
hwreal : ComplexEmbedding.IsReal w.embedding
f : K →+* ℂ := w.embedding
hζ' : IsPrimitiveRoot (M := ℂ) ((f : (a : K) → ℂ) ζ : ℂ) k
⊢ Complex.im ((f : (a : K) → ℂ) ζ) = 0
have rw : Eq (α := ℂ) ((ComplexEmbedding.conjugate f : (a : K) → ℂ) ζ : ℂ) ((f : (a : K) → ℂ) ζ : ℂ)
Tactic: rw [← Complex.conj_eq_iff_im, ← NumberField.ComplexEmbedding.conjugate_coe_eq]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
x : { w : InfinitePlace K // w.IsReal }
w : InfinitePlace K
hwreal : ComplexEmbedding.IsReal w.embedding
f : K →+* ℂ := w.embedding
hζ' : IsPrimitiveRoot (M := ℂ) ((f : (a : K) → ℂ) ζ : ℂ) k
⊢ False
have him : Complex.im ((f : (a : K) → ℂ) ζ) = 0
Tactic: have him : (f ζ).im = 0 := by
    rw [← Complex.conj_eq_iff_im, ← NumberField.ComplexEmbedding.conjugate_coe_eq]
    congr
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
x : { w : InfinitePlace K // w.IsReal }
w : InfinitePlace K
hwreal : ComplexEmbedding.IsReal w.embedding
f : K →+* ℂ := w.embedding
hζ' : IsPrimitiveRoot (M := ℂ) ((f : (a : K) → ℂ) ζ : ℂ) k
him : Complex.im ((f : (a : K) → ℂ) ζ) = 0
⊢ Complex.re ((f : (a : K) → ℂ) ζ) = 1 ∨ Complex.re ((f : (a : K) → ℂ) ζ) = -1
have him₁ : |Complex.re ((f : (a : K) → ℂ) ζ)| = norm (E := ℂ) ((f : (a : K) → ℂ) ζ : ℂ)
Tactic: rw [← Complex.abs_re_eq_norm] at him
Cleared hypotheses: him
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
x : { w : InfinitePlace K // w.IsReal }
w : InfinitePlace K
hwreal : ComplexEmbedding.IsReal w.embedding
f : K →+* ℂ := w.embedding
hζ' : IsPrimitiveRoot (M := ℂ) ((f : (a : K) → ℂ) ζ : ℂ) k
him : |Complex.re ((f : (a : K) → ℂ) ζ)| = norm (E := ℂ) ((f : (a : K) → ℂ) ζ : ℂ)
⊢ Complex.re ((f : (a : K) → ℂ) ζ) = 1 ∨ Complex.re ((f : (a : K) → ℂ) ζ) = -1
have this : norm (E := ℂ) ((f : (a : K) → ℂ) ζ : ℂ) = 1
Tactic: have := Complex.norm_eq_one_of_pow_eq_one hζ'.pow_eq_one (by lia)
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
x : { w : InfinitePlace K // w.IsReal }
w : InfinitePlace K
hwreal : ComplexEmbedding.IsReal w.embedding
f : K →+* ℂ := w.embedding
hζ' : IsPrimitiveRoot (M := ℂ) ((f : (a : K) → ℂ) ζ : ℂ) k
him : |Complex.re ((f : (a : K) → ℂ) ζ)| = norm (E := ℂ) ((f : (a : K) → ℂ) ζ : ℂ)
this : norm (E := ℂ) ((f : (a : K) → ℂ) ζ : ℂ) = 1
⊢ Complex.re ((f : (a : K) → ℂ) ζ) = 1 ∨ Complex.re ((f : (a : K) → ℂ) ζ) = -1
have rwa : ∀ (this : |Complex.re ((f : (a : K) → ℂ) ζ)| = 1),
  Complex.re ((f : (a : K) → ℂ) ζ) = 1 ∨ Complex.re ((f : (a : K) → ℂ) ζ) = -1
Tactic: rwa [← him, ← abs_one, abs_eq_abs] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
x : { w : InfinitePlace K // w.IsReal }
w : InfinitePlace K
hwreal : ComplexEmbedding.IsReal w.embedding
f : K →+* ℂ := w.embedding
hζ' : IsPrimitiveRoot (M := ℂ) ((f : (a : K) → ℂ) ζ : ℂ) k
him : |Complex.re ((f : (a : K) → ℂ) ζ)| = norm (E := ℂ) ((f : (a : K) → ℂ) ζ : ℂ)
this : norm (E := ℂ) ((f : (a : K) → ℂ) ζ : ℂ) = 1
rwa :
  ∀ (this : |Complex.re ((f : (a : K) → ℂ) ζ)| = 1),
    Complex.re ((f : (a : K) → ℂ) ζ) = 1 ∨ Complex.re ((f : (a : K) → ℂ) ζ) = -1
⊢ Complex.re ((f : (a : K) → ℂ) ζ) = 1 ∨ Complex.re ((f : (a : K) → ℂ) ζ) = -1
have rwa₁ : ∀ (this : |Complex.re ((f : (a : K) → ℂ) ζ)| = |1|),
  Complex.re ((f : (a : K) → ℂ) ζ) = 1 ∨ Complex.re ((f : (a : K) → ℂ) ζ) = -1
Tactic: rwa [← him, ← abs_one, abs_eq_abs] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
ζ : K
k : ℕ
hk : 2 < k
hζ : IsPrimitiveRoot ζ k
x : { w : InfinitePlace K // w.IsReal }
w : InfinitePlace K
hwreal : ComplexEmbedding.IsReal w.embedding
f : K →+* ℂ := w.embedding
hζ' : IsPrimitiveRoot (M := ℂ) ((f : (a : K) → ℂ) ζ : ℂ) k
him : |Complex.re ((f : (a : K) → ℂ) ζ)| = norm (E := ℂ) ((f : (a : K) → ℂ) ζ : ℂ)
this : norm (E := ℂ) ((f : (a : K) → ℂ) ζ : ℂ) = 1
rwa :
  ∀ (this : |Complex.re ((f : (a : K) → ℂ) ζ)| = 1),
    Complex.re ((f : (a : K) → ℂ) ζ) = 1 ∨ Complex.re ((f : (a : K) → ℂ) ζ) = -1
rwa₁ :
  ∀ (this : |Complex.re ((f : (a : K) → ℂ) ζ)| = |1|),
    Complex.re ((f : (a : K) → ℂ) ζ) = 1 ∨ Complex.re ((f : (a : K) → ℂ) ζ) = -1
⊢ Complex.re ((f : (a : K) → ℂ) ζ) = 1 ∨ Complex.re ((f : (a : K) → ℂ) ζ) = -1
have rwa₂ : ∀ (this : Complex.re ((f : (a : K) → ℂ) ζ) = 1 ∨ Complex.re ((f : (a : K) → ℂ) ζ) = -1),
  Complex.re ((f : (a : K) → ℂ) ζ) = 1 ∨ Complex.re ((f : (a : K) → ℂ) ζ) = -1
Tactic: rwa [← him, ← abs_one, abs_eq_abs] at this
Cleared hypotheses: this