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
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → LT.lt (α := ℝ) ((w' : (a : K) → ℝ) (↑x : K) : ℝ) (1 : ℝ)
h₃ : w.IsReal ∨ |Complex.re ((w.embedding : (a : K) → ℂ) (↑x : K))| < 1
⊢ Eq (α := Subalgebra ℚ K) ℚ[(↑x : K)] ⊤
have rw : Eq (α := Subalgebra ℚ K) (IntermediateField.toSubalgebra ℚ⟮(↑x : K)⟯) ⊤
Tactic: rw [← IntermediateField.adjoin_simple_toSubalgebra_of_isAlgebraic (IsAlgebraic.of_finite ℚ _)]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
⊢ card.{u_1} (InfinitePlace K) = nrRealPlaces K + nrComplexPlaces K
have h : card.{u_1} (InfinitePlace K) = card.{u_1} { x : InfinitePlace K // x.IsReal ∨ x.IsComplex }
Tactic: convert!
    Fintype.card_subtype_or_disjoint (IsReal (K := K)) (IsComplex (K := K))
      (disjoint_isReal_isComplex K) using 1
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
⊢ card.{u_1} (InfinitePlace K) = nrRealPlaces K + nrComplexPlaces K
have h : card.{u_1} (InfinitePlace K) = card.{u_1} { x : InfinitePlace K // x.IsReal ∨ x.IsComplex }
Tactic: classical
  convert!
    Fintype.card_subtype_or_disjoint (IsReal (K := K)) (IsComplex (K := K))
      (disjoint_isReal_isComplex K) using 1
  exact (Fintype.card_of_subtype _ (fun w ↦ ⟨fun _ ↦ isReal_or_isComplex w, fun _ ↦ by simp⟩)).symm
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
this :
  ∀ (w : { w : InfinitePlace K // w.IsComplex }),
    Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
        {φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex φ = w} =
      2
⊢ card.{u_1} { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } = 2 * nrComplexPlaces K
have rw : ∑ j : { w : InfinitePlace K // w.IsComplex },
    ∑ i : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } with mkComplex i = j, 1 =
  2 * nrComplexPlaces K
Tactic: rw [Fintype.card, Finset.card_eq_sum_ones, ← Finset.sum_fiberwise _ (fun φ => mkComplex φ)]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
this :
  ∀ (w : { w : InfinitePlace K // w.IsComplex }),
    Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
        {φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex φ = w} =
      2
⊢ ∑ j : { w : InfinitePlace K // w.IsComplex },
      ∑ i : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } with mkComplex i = j, 1 =
    2 * nrComplexPlaces K
have simp_rw : ∑ x : { w : InfinitePlace K // w.IsComplex },
    HSMul.hSMul (β := ℕ)
      (Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
        {i : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex i = x})
      1 =
  2 * nrComplexPlaces K
Tactic: simp_rw [Finset.sum_const, this, smul_eq_mul, mul_one, Fintype.card, Finset.card_eq_sum_ones,
      Finset.mul_sum, Finset.sum_const, smul_eq_mul, mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
this :
  ∀ (w : { w : InfinitePlace K // w.IsComplex }),
    Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
        {φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex φ = w} =
      2
simp_rw :
  ∑ x : { w : InfinitePlace K // w.IsComplex },
      HSMul.hSMul (β := ℕ)
        (Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
          {i : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex i = x})
        1 =
    2 * nrComplexPlaces K
⊢ ∑ j : { w : InfinitePlace K // w.IsComplex },
      ∑ i : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } with mkComplex i = j, 1 =
    2 * nrComplexPlaces K
have simp_rw₁ : ∑ x : { w : InfinitePlace K // w.IsComplex }, HSMul.hSMul (α := ℕ) (β := ℕ) 2 1 = 2 * nrComplexPlaces K
Tactic: simp_rw [Finset.sum_const, this, smul_eq_mul, mul_one, Fintype.card, Finset.card_eq_sum_ones,
      Finset.mul_sum, Finset.sum_const, smul_eq_mul, mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
this :
  ∀ (w : { w : InfinitePlace K // w.IsComplex }),
    Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
        {φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex φ = w} =
      2
simp_rw :
  ∑ x : { w : InfinitePlace K // w.IsComplex },
      HSMul.hSMul (β := ℕ)
        (Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
          {i : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex i = x})
        1 =
    2 * nrComplexPlaces K
simp_rw₁ : ∑ x : { w : InfinitePlace K // w.IsComplex }, HSMul.hSMul (α := ℕ) (β := ℕ) 2 1 = 2 * nrComplexPlaces K
⊢ ∑ j : { w : InfinitePlace K // w.IsComplex },
      ∑ i : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } with mkComplex i = j, 1 =
    2 * nrComplexPlaces K
have simp_rw₂ : ∑ x : { w : InfinitePlace K // w.IsComplex }, HMul.hMul (α := ℕ) 2 1 = 2 * nrComplexPlaces K
Tactic: simp_rw [Finset.sum_const, this, smul_eq_mul, mul_one, Fintype.card, Finset.card_eq_sum_ones,
      Finset.mul_sum, Finset.sum_const, smul_eq_mul, mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
this :
  ∀ (w : { w : InfinitePlace K // w.IsComplex }),
    Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
        {φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex φ = w} =
      2
simp_rw :
  ∑ x : { w : InfinitePlace K // w.IsComplex },
      HSMul.hSMul (β := ℕ)
        (Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
          {i : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex i = x})
        1 =
    2 * nrComplexPlaces K
simp_rw₁ : ∑ x : { w : InfinitePlace K // w.IsComplex }, HSMul.hSMul (α := ℕ) (β := ℕ) 2 1 = 2 * nrComplexPlaces K
simp_rw₂ : ∑ x : { w : InfinitePlace K // w.IsComplex }, HMul.hMul (α := ℕ) 2 1 = 2 * nrComplexPlaces K
simp_rw₃ : ∑ x : { w : InfinitePlace K // w.IsComplex }, 2 = 2 * nrComplexPlaces K
⊢ ∑ j : { w : InfinitePlace K // w.IsComplex },
      ∑ i : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } with mkComplex i = j, 1 =
    2 * nrComplexPlaces K
have simp_rw₄ : ∑ x : { w : InfinitePlace K // w.IsComplex }, 2 =
  2 * Finset.card.{u_1} (α := { w : InfinitePlace K // w.IsComplex }) Finset.univ
Tactic: simp_rw [Finset.sum_const, this, smul_eq_mul, mul_one, Fintype.card, Finset.card_eq_sum_ones,
      Finset.mul_sum, Finset.sum_const, smul_eq_mul, mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
this :
  ∀ (w : { w : InfinitePlace K // w.IsComplex }),
    Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
        {φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex φ = w} =
      2
simp_rw :
  ∑ x : { w : InfinitePlace K // w.IsComplex },
      HSMul.hSMul (β := ℕ)
        (Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
          {i : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex i = x})
        1 =
    2 * nrComplexPlaces K
simp_rw₁ : ∑ x : { w : InfinitePlace K // w.IsComplex }, HSMul.hSMul (α := ℕ) (β := ℕ) 2 1 = 2 * nrComplexPlaces K
simp_rw₂ : ∑ x : { w : InfinitePlace K // w.IsComplex }, HMul.hMul (α := ℕ) 2 1 = 2 * nrComplexPlaces K
simp_rw₃ : ∑ x : { w : InfinitePlace K // w.IsComplex }, 2 = 2 * nrComplexPlaces K
simp_rw₄ :
  ∑ x : { w : InfinitePlace K // w.IsComplex }, 2 =
    2 * Finset.card.{u_1} (α := { w : InfinitePlace K // w.IsComplex }) Finset.univ
⊢ ∑ j : { w : InfinitePlace K // w.IsComplex },
      ∑ i : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } with mkComplex i = j, 1 =
    2 * nrComplexPlaces K
have simp_rw₅ : ∑ x : { w : InfinitePlace K // w.IsComplex }, 2 = HMul.hMul (α := ℕ) 2 (∑ x : { w : InfinitePlace K // w.IsComplex }, 1)
Tactic: simp_rw [Finset.sum_const, this, smul_eq_mul, mul_one, Fintype.card, Finset.card_eq_sum_ones,
      Finset.mul_sum, Finset.sum_const, smul_eq_mul, mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
this :
  ∀ (w : { w : InfinitePlace K // w.IsComplex }),
    Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
        {φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex φ = w} =
      2
simp_rw :
  ∑ x : { w : InfinitePlace K // w.IsComplex },
      HSMul.hSMul (β := ℕ)
        (Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
          {i : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex i = x})
        1 =
    2 * nrComplexPlaces K
simp_rw₁ : ∑ x : { w : InfinitePlace K // w.IsComplex }, HSMul.hSMul (α := ℕ) (β := ℕ) 2 1 = 2 * nrComplexPlaces K
simp_rw₂ : ∑ x : { w : InfinitePlace K // w.IsComplex }, HMul.hMul (α := ℕ) 2 1 = 2 * nrComplexPlaces K
simp_rw₃ : ∑ x : { w : InfinitePlace K // w.IsComplex }, 2 = 2 * nrComplexPlaces K
simp_rw₄ :
  ∑ x : { w : InfinitePlace K // w.IsComplex }, 2 =
    2 * Finset.card.{u_1} (α := { w : InfinitePlace K // w.IsComplex }) Finset.univ
simp_rw₅ :
  ∑ x : { w : InfinitePlace K // w.IsComplex }, 2 =
    HMul.hMul (α := ℕ) 2 (∑ x : { w : InfinitePlace K // w.IsComplex }, 1)
⊢ ∑ j : { w : InfinitePlace K // w.IsComplex },
      ∑ i : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } with mkComplex i = j, 1 =
    2 * nrComplexPlaces K
have simp_rw₆ : Eq (α := ℕ) (∑ x : { w : InfinitePlace K // w.IsComplex }, 2)
  (∑ i : { w : InfinitePlace K // w.IsComplex }, HMul.hMul (α := ℕ) 2 1)
Tactic: simp_rw [Finset.sum_const, this, smul_eq_mul, mul_one, Fintype.card, Finset.card_eq_sum_ones,
      Finset.mul_sum, Finset.sum_const, smul_eq_mul, mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
this :
  ∀ (w : { w : InfinitePlace K // w.IsComplex }),
    Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
        {φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex φ = w} =
      2
simp_rw :
  ∑ x : { w : InfinitePlace K // w.IsComplex },
      HSMul.hSMul (β := ℕ)
        (Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
          {i : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex i = x})
        1 =
    2 * nrComplexPlaces K
simp_rw₁ : ∑ x : { w : InfinitePlace K // w.IsComplex }, HSMul.hSMul (α := ℕ) (β := ℕ) 2 1 = 2 * nrComplexPlaces K
simp_rw₂ : ∑ x : { w : InfinitePlace K // w.IsComplex }, HMul.hMul (α := ℕ) 2 1 = 2 * nrComplexPlaces K
simp_rw₃ : ∑ x : { w : InfinitePlace K // w.IsComplex }, 2 = 2 * nrComplexPlaces K
simp_rw₄ :
  ∑ x : { w : InfinitePlace K // w.IsComplex }, 2 =
    2 * Finset.card.{u_1} (α := { w : InfinitePlace K // w.IsComplex }) Finset.univ
simp_rw₅ :
  ∑ x : { w : InfinitePlace K // w.IsComplex }, 2 =
    HMul.hMul (α := ℕ) 2 (∑ x : { w : InfinitePlace K // w.IsComplex }, 1)
simp_rw₆ :
  Eq (α := ℕ) (∑ x : { w : InfinitePlace K // w.IsComplex }, 2)
    (∑ i : { w : InfinitePlace K // w.IsComplex }, HMul.hMul (α := ℕ) 2 1)
⊢ ∑ j : { w : InfinitePlace K // w.IsComplex },
      ∑ i : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } with mkComplex i = j, 1 =
    2 * nrComplexPlaces K
have simp_rw₇ : HSMul.hSMul (β := ℕ) (Finset.card.{u_1} (α := { w : InfinitePlace K // w.IsComplex }) Finset.univ) 2 =
  Finset.card.{u_1} (α := { w : InfinitePlace K // w.IsComplex }) Finset.univ • HMul.hMul (α := ℕ) 2 1
Tactic: simp_rw [Finset.sum_const, this, smul_eq_mul, mul_one, Fintype.card, Finset.card_eq_sum_ones,
      Finset.mul_sum, Finset.sum_const, smul_eq_mul, mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
this :
  ∀ (w : { w : InfinitePlace K // w.IsComplex }),
    Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
        {φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex φ = w} =
      2
simp_rw :
  ∑ x : { w : InfinitePlace K // w.IsComplex },
      HSMul.hSMul (β := ℕ)
        (Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
          {i : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex i = x})
        1 =
    2 * nrComplexPlaces K
simp_rw₁ : ∑ x : { w : InfinitePlace K // w.IsComplex }, HSMul.hSMul (α := ℕ) (β := ℕ) 2 1 = 2 * nrComplexPlaces K
simp_rw₂ : ∑ x : { w : InfinitePlace K // w.IsComplex }, HMul.hMul (α := ℕ) 2 1 = 2 * nrComplexPlaces K
simp_rw₃ : ∑ x : { w : InfinitePlace K // w.IsComplex }, 2 = 2 * nrComplexPlaces K
simp_rw₄ :
  ∑ x : { w : InfinitePlace K // w.IsComplex }, 2 =
    2 * Finset.card.{u_1} (α := { w : InfinitePlace K // w.IsComplex }) Finset.univ
simp_rw₅ :
  ∑ x : { w : InfinitePlace K // w.IsComplex }, 2 =
    HMul.hMul (α := ℕ) 2 (∑ x : { w : InfinitePlace K // w.IsComplex }, 1)
simp_rw₆ :
  Eq (α := ℕ) (∑ x : { w : InfinitePlace K // w.IsComplex }, 2)
    (∑ i : { w : InfinitePlace K // w.IsComplex }, HMul.hMul (α := ℕ) 2 1)
simp_rw₇ :
  HSMul.hSMul (β := ℕ) (Finset.card.{u_1} (α := { w : InfinitePlace K // w.IsComplex }) Finset.univ) 2 =
    Finset.card.{u_1} (α := { w : InfinitePlace K // w.IsComplex }) Finset.univ • HMul.hMul (α := ℕ) 2 1
⊢ ∑ j : { w : InfinitePlace K // w.IsComplex },
      ∑ i : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } with mkComplex i = j, 1 =
    2 * nrComplexPlaces K
have simp_rw₈ : Finset.card.{u_1} (α := { w : InfinitePlace K // w.IsComplex }) Finset.univ * 2 =
  Finset.card.{u_1} (α := { w : InfinitePlace K // w.IsComplex }) Finset.univ * HMul.hMul (α := ℕ) 2 1
Tactic: simp_rw [Finset.sum_const, this, smul_eq_mul, mul_one, Fintype.card, Finset.card_eq_sum_ones,
      Finset.mul_sum, Finset.sum_const, smul_eq_mul, mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.InfinitePlace.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
⊢ card.{u_1} { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } = 2 * nrComplexPlaces K
have suffices : ∀ (w : { w : InfinitePlace K // w.IsComplex }),
  Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
      {φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } | mkComplex φ = w} =
    2
Tactic: suffices ∀ w : { w : InfinitePlace K // IsComplex w },
     #{φ : {φ //¬ ComplexEmbedding.IsReal φ} | mkComplex φ = w} = 2 by
    rw [Fintype.card, Finset.card_eq_sum_ones, ← Finset.sum_fiberwise _ (fun φ => mkComplex φ)]
    simp_rw [Finset.sum_const, this, smul_eq_mul, mul_one, Fintype.card, Finset.card_eq_sum_ones,
      Finset.mul_sum, Finset.sum_const, smul_eq_mul, mul_one]
-- 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
⊢ Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
      {φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } |
        mkComplex φ = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw} =
    2
have h : Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
    {φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } |
      mkComplex φ = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw} =
  Finset.card.{u_1} (α := K →+* ℂ) {φ : K →+* ℂ | mk φ = w}
Tactic: convert! card_filter_mk_eq w
-- 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 :
  Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
      {φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } |
        mkComplex φ = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw} =
    Finset.card.{u_1} (α := K →+* ℂ) {φ : K →+* ℂ | mk φ = w}
⊢ Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
      {φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } |
        mkComplex φ = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw} =
    2
have h₁ : 2 = w.mult
Tactic: convert! card_filter_mk_eq w
-- 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
⊢ Finset.card.{u_1} (α := { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ })
      {φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } |
        mkComplex φ = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw} =
    Finset.card.{u_1} (α := K →+* ℂ) {φ : K →+* ℂ | mk φ = w}
have h : card.{u_1}
    { x : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } //
      mkComplex x = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw } =
  card.{u_1} { x : K →+* ℂ // mk x = w }
Tactic: rw [← Fintype.card_subtype, ← Fintype.card_subtype]
-- 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
⊢ card.{u_1}
      { x : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } //
        mkComplex x = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw } =
    card.{u_1} { x : K →+* ℂ // mk x = w }
have h : { 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 }
Tactic: refine Fintype.card_congr (Equiv.ofBijective ?_ ⟨fun _ _ h => ?_, fun ⟨φ, hφ⟩ => ?_⟩)
-- 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 :
  { 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 }
⊢ card.{u_1}
      { x : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } //
        mkComplex x = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw } =
    card.{u_1} { x : K →+* ℂ // mk x = w }
have h₁ : ∀
  (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) sorry :
        { 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) sorry :
        { x : K →+* ℂ // mk x = w })),
  x = x_1
Tactic: refine Fintype.card_congr (Equiv.ofBijective ?_ ⟨fun _ _ h => ?_, fun ⟨φ, hφ⟩ => ?_⟩)
-- 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 :
  { 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 }
h₁ :
  ∀
    (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 })),
    x = x_1
⊢ card.{u_1}
      { x : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } //
        mkComplex x = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw } =
    card.{u_1} { x : K →+* ℂ // mk x = w }
have h₂ : ∀ (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) sorry :
        { x : K →+* ℂ // mk x = w })
      (Subtype.mk (p := fun (x : K →+* ℂ) ↦ mk x = w) φ hφ : { x : K →+* ℂ // mk x = w })
Tactic: refine Fintype.card_congr (Equiv.ofBijective ?_ ⟨fun _ _ h => ?_, fun ⟨φ, hφ⟩ => ?_⟩)
-- 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 : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } //
    mkComplex x = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw }
φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ }
hφ : mkComplex φ = Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw
⊢ mk (↑φ : K →+* ℂ) = w
have rwa : ∀
  (hφ :
    (↑(mkComplex φ) : InfinitePlace K) =
      (↑(Subtype.mk (p := fun (w : InfinitePlace K) ↦ w.IsComplex) w hw) : InfinitePlace K)),
  mk (↑φ : K →+* ℂ) = w
Tactic: rwa [Subtype.ext_iff] at hφ
Cleared hypotheses: hφ
-- 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 })
⊢ x = x_1
have h₁ : ∀
  (h :
    (↑(↑x : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ }) : K →+* ℂ) =
      (↑(↑x_1 : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ }) : K →+* ℂ)),
  x = x_1
Tactic: rwa [Subtype.mk_eq_mk, ← Subtype.ext_iff, ← Subtype.ext_iff] at h
Cleared hypotheses: h