awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
x : (𝓞 K)ˣ
h :
  Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x ↔
    Eq (α := Multiplicative (logSpace K))
      ((((AddMonoidHom.toMultiplicativeRight :
                  (a : Additive (𝓞 K)ˣ →+ logSpace K) → (𝓞 K)ˣ →* Multiplicative (logSpace K))
                (logEmbedding K) :
              (𝓞 K)ˣ →* Multiplicative (logSpace K)) :
            (a : (𝓞 K)ˣ) → Multiplicative (logSpace K))
          x :
        Multiplicative (logSpace K))
      (1 : Multiplicative (logSpace K))
h₁ :
  Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x ↔
    Eq (α := Multiplicative (logSpace K))
      ((Multiplicative.ofAdd : (a : logSpace K) → Multiplicative (logSpace K))
          ((logEmbedding K : (a : Additive (𝓞 K)ˣ) → logSpace K)
              ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) x : Additive (𝓞 K)ˣ) :
            logSpace K) :
        Multiplicative (logSpace K))
      (1 : Multiplicative (logSpace K))
⊢ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x ↔
    Membership.mem (γ := Subgroup (𝓞 K)ˣ)
      (MonoidHom.ker (M := Multiplicative (logSpace K))
        ((AddMonoidHom.toMultiplicativeRight :
            (a : Additive (𝓞 K)ˣ →+ logSpace K) → (𝓞 K)ˣ →* Multiplicative (logSpace K))
          (logEmbedding K)))
      x
have h₂ : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x ↔
  Eq (α := logSpace K)
    ((logEmbedding K : (a : Additive (𝓞 K)ˣ) → logSpace K)
        ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) x : Additive (𝓞 K)ˣ) :
      logSpace K)
    (0 : logSpace K)
Tactic: rw [MonoidHom.mem_ker, AddMonoidHom.toMultiplicativeRight_apply_apply, ofAdd_eq_one,
          ← logEmbedding_eq_zero_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
x : (𝓞 K)ˣ
h :
  Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x ↔
    Eq (α := Multiplicative (logSpace K))
      ((((AddMonoidHom.toMultiplicativeRight :
                  (a : Additive (𝓞 K)ˣ →+ logSpace K) → (𝓞 K)ˣ →* Multiplicative (logSpace K))
                (logEmbedding K) :
              (𝓞 K)ˣ →* Multiplicative (logSpace K)) :
            (a : (𝓞 K)ˣ) → Multiplicative (logSpace K))
          x :
        Multiplicative (logSpace K))
      (1 : Multiplicative (logSpace K))
h₁ :
  Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x ↔
    Eq (α := Multiplicative (logSpace K))
      ((Multiplicative.ofAdd : (a : logSpace K) → Multiplicative (logSpace K))
          ((logEmbedding K : (a : Additive (𝓞 K)ˣ) → logSpace K)
              ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) x : Additive (𝓞 K)ˣ) :
            logSpace K) :
        Multiplicative (logSpace K))
      (1 : Multiplicative (logSpace K))
h₂ :
  Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x ↔
    Eq (α := logSpace K)
      ((logEmbedding K : (a : Additive (𝓞 K)ˣ) → logSpace K)
          ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) x : Additive (𝓞 K)ˣ) :
        logSpace K)
      (0 : logSpace K)
⊢ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x ↔
    Membership.mem (γ := Subgroup (𝓞 K)ˣ)
      (MonoidHom.ker (M := Multiplicative (logSpace K))
        ((AddMonoidHom.toMultiplicativeRight :
            (a : Additive (𝓞 K)ˣ →+ logSpace K) → (𝓞 K)ˣ →* Multiplicative (logSpace K))
          (logEmbedding K)))
      x
have h₃ : Eq (α := logSpace K)
    ((logEmbedding K : (a : Additive (𝓞 K)ˣ) → logSpace K)
        ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) x : Additive (𝓞 K)ˣ) :
      logSpace K)
    (0 : logSpace K) ↔
  Eq (α := logSpace K)
    ((logEmbedding K : (a : Additive (𝓞 K)ˣ) → logSpace K)
        ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) x : Additive (𝓞 K)ˣ) :
      logSpace K)
    (0 : logSpace K)
Tactic: rw [MonoidHom.mem_ker, AddMonoidHom.toMultiplicativeRight_apply_apply, ofAdd_eq_one,
          ← logEmbedding_eq_zero_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
⊢ Module.Finite.{0, u_1} ℤ (Additive (𝓞 K)ˣ)
have rw : Submodule.FG.{0, u_1} (R := ℤ) (M := Additive (𝓞 K)ˣ) ⊤
Tactic: rw [Module.finite_def]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
⊢ Monoid.FG.{u_1} (𝓞 K)ˣ
have rw : Module.Finite.{0, u_1} ℤ (Additive (𝓞 K)ˣ)
Tactic: rw [Monoid.fg_iff_add_fg, ← AddGroup.fg_iff_addMonoid_fg, ← Module.Finite.iff_addGroup_fg]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
x ζ : (𝓞 K)ˣ
f : Fin (rank K) → ℤ
hζ : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) ζ
h : x = ζ * ∏ i : Fin (rank K), fundSystem K i ^ f i
⊢ f = sorry
have suffices : Eq (α := Additive (𝓞 K)ˣ) ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) x : Additive (𝓞 K)ˣ)
  (∑ i : Fin (rank K), HSMul.hSMul (β := Additive (𝓞 K)ˣ) (f i) sorry : Additive (𝓞 K)ˣ)
Tactic: suffices Additive.ofMul ↑x = ∑ i, (f i) • (basisModTorsion K i) by
    rw [← (basisModTorsion K).repr_sum_self f, ← this]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
x ζ : (𝓞 K)ˣ
f : Fin (rank K) → ℤ
hζ : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) ζ
h : x = ζ * ∏ i : Fin (rank K), fundSystem K i ^ f i
⊢ Eq (α := Additive (𝓞 K)ˣ) ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) x : Additive (𝓞 K)ˣ)
    (∑ i : Fin (rank K),
        HSMul.hSMul (β := Additive (𝓞 K)ˣ) (f i)
          ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) (fundSystem K i) : Additive (𝓞 K)ˣ) :
      Additive (𝓞 K)ˣ)
have rw : Eq (α := Additive (𝓞 K)ˣ)
  ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) (ζ * ∏ i : Fin (rank K), fundSystem K i ^ f i) : Additive (𝓞 K)ˣ)
  (∑ i : Fin (rank K),
      HSMul.hSMul (β := Additive (𝓞 K)ˣ) (f i)
        ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) (fundSystem K i) : Additive (𝓞 K)ˣ) :
    Additive (𝓞 K)ˣ)
Tactic: rw [h, QuotientGroup.mk_mul, (QuotientGroup.eq_one_iff _).mpr hζ, one_mul,
            QuotientGroup.mk_prod, ofMul_prod]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
x ζ : (𝓞 K)ˣ
f : Fin (rank K) → ℤ
hζ : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) ζ
h : x = ζ * ∏ i : Fin (rank K), fundSystem K i ^ f i
⊢ Eq.{u_1 + 1} (α := Additive (𝓞 K)ˣ)
    (∑ i : Fin (rank K),
      HSMul.hSMul (β := Additive (𝓞 K)ˣ) (f i)
        ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) (fundSystem K i) : Additive (𝓞 K)ˣ))
    (∑ i : Fin (rank K), HSMul.hSMul (β := Additive (𝓞 K)ˣ) (f i) sorry)
have simp_rw : Eq.{u_1 + 1} (α := Additive (𝓞 K)ˣ)
  (∑ x : Fin (rank K),
    HSMul.hSMul (β := Additive (𝓞 K)ˣ) (f x) ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) (sorry K x)))
  (∑ i : Fin (rank K), HSMul.hSMul (β := Additive (𝓞 K)ˣ) (f i) sorry)
Tactic: simp_rw [fundSystem, QuotientGroup.out_eq', ofMul_toMul]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
x ζ : (𝓞 K)ˣ
f : Fin (rank K) → ℤ
hζ : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) ζ
h : x = ζ * ∏ i : Fin (rank K), fundSystem K i ^ f i
⊢ Eq (α := Additive (𝓞 K)ˣ) ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) x : Additive (𝓞 K)ˣ)
    (∑ i : Fin (rank K), HSMul.hSMul (β := Additive (𝓞 K)ˣ) (f i) sorry : Additive (𝓞 K)ˣ)
have _ : Eq (α := Additive (𝓞 K)ˣ) ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) x : Additive (𝓞 K)ˣ)
  (∑ i : Fin (rank K),
      HSMul.hSMul (β := Additive (𝓞 K)ˣ) (f i)
        ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) (fundSystem K i) : Additive (𝓞 K)ˣ) :
    Additive (𝓞 K)ˣ)
Tactic: calc
    Additive.ofMul ↑x
    _ = ∑ i, (f i) • Additive.ofMul ↑(fundSystem K i) := by
          rw [h, QuotientGroup.mk_mul, (QuotientGroup.eq_one_iff _).mpr hζ, one_mul,
            QuotientGroup.mk_prod, ofMul_prod]; rfl
    _ = ∑ i, (f i) • (basisModTorsion K i) := by
          simp_rw [fundSystem, QuotientGroup.out_eq', ofMul_toMul]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
x ζ : (𝓞 K)ˣ
f : Fin (rank K) → ℤ
hζ : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) ζ
h : x = ζ * ∏ i : Fin (rank K), fundSystem K i ^ f i
_ :
  Eq (α := Additive (𝓞 K)ˣ) ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) x : Additive (𝓞 K)ˣ)
    (∑ i : Fin (rank K),
        HSMul.hSMul (β := Additive (𝓞 K)ˣ) (f i)
          ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) (fundSystem K i) : Additive (𝓞 K)ˣ) :
      Additive (𝓞 K)ˣ)
⊢ Eq (α := Additive (𝓞 K)ˣ) ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) x : Additive (𝓞 K)ˣ)
    (∑ i : Fin (rank K), HSMul.hSMul (β := Additive (𝓞 K)ˣ) (f i) sorry : Additive (𝓞 K)ˣ)
have _₁ : Eq.{u_1 + 1} (α := Additive (𝓞 K)ˣ)
  (∑ i : Fin (rank K),
    HSMul.hSMul (β := Additive (𝓞 K)ˣ) (f i)
      ((Additive.ofMul : (a : (𝓞 K)ˣ) → Additive (𝓞 K)ˣ) (fundSystem K i) : Additive (𝓞 K)ˣ))
  (∑ i : Fin (rank K), HSMul.hSMul (β := Additive (𝓞 K)ˣ) (f i) sorry)
Tactic: calc
    Additive.ofMul ↑x
    _ = ∑ i, (f i) • Additive.ofMul ↑(fundSystem K i) := by
          rw [h, QuotientGroup.mk_mul, (QuotientGroup.eq_one_iff _).mpr hζ, one_mul,
            QuotientGroup.mk_prod, ofMul_prod]; rfl
    _ = ∑ i, (f i) • (basisModTorsion K i) := by
          simp_rw [fundSystem, QuotientGroup.out_eq', ofMul_toMul]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
x : (𝓞 K)ˣ
ζ : (𝓞 K)ˣ := x * (∏ i : Fin (rank K), HPow.hPow (β := ℕ) (fundSystem K i) sorry)⁻¹
⊢ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) ζ
have rw : HAdd.hAdd (α := Additive (HasQuotient.Quotient (𝓞 K)ˣ (B := Subgroup (𝓞 K)ˣ) (torsion K))) (β :=
    Additive (HasQuotient.Quotient (𝓞 K)ˣ (B := Subgroup (𝓞 K)ˣ) (torsion K)))
    ((Additive.ofMul : (a : (𝓞 K)ˣ ⧸ torsion K) → Additive ((𝓞 K)ˣ ⧸ torsion K)) (↑x : (𝓞 K)ˣ ⧸ torsion K) :
      Additive (HasQuotient.Quotient (𝓞 K)ˣ (B := Subgroup (𝓞 K)ˣ) (torsion K)))
    (-∑ i : Fin (rank K),
          (Additive.ofMul : (a : (𝓞 K)ˣ ⧸ torsion K) → Additive ((𝓞 K)ˣ ⧸ torsion K))
            (↑(HPow.hPow (β := ℕ) (fundSystem K i) sorry) : (𝓞 K)ˣ ⧸ torsion K) :
      Additive (HasQuotient.Quotient (𝓞 K)ˣ (B := Subgroup (𝓞 K)ˣ) (torsion K))) =
  0
Tactic: rw [← QuotientGroup.eq_one_iff, QuotientGroup.mk_mul, QuotientGroup.mk_inv, ← ofMul_eq_zero,
      ofMul_mul, ofMul_inv, QuotientGroup.mk_prod, ofMul_prod]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
x : (𝓞 K)ˣ
ζ : (𝓞 K)ˣ := x * (∏ i : Fin (rank K), HPow.hPow (β := ℕ) (fundSystem K i) sorry)⁻¹
⊢ ∃! ζe : ↥(torsion K) × (Fin (rank K) → ℤ), x = (↑ζe.1 : (𝓞 K)ˣ) * ∏ i : Fin (rank K), fundSystem K i ^ ζe.2 i
have h_tors : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) ζ
Tactic: have h_tors : ζ ∈ torsion K := by
    rw [← QuotientGroup.eq_one_iff, QuotientGroup.mk_mul, QuotientGroup.mk_inv, ← ofMul_eq_zero,
      ofMul_mul, ofMul_inv, QuotientGroup.mk_prod, ofMul_prod]
    simp_rw [QuotientGroup.mk_zpow, ofMul_zpow, fundSystem, QuotientGroup.out_eq']
    rw [add_eq_zero_iff_eq_neg, neg_neg]
    exact ((basisModTorsion K).sum_repr (Additive.ofMul ↑x)).symm
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
x : (𝓞 K)ˣ
ζ : (𝓞 K)ˣ := x * (∏ i : Fin (rank K), HPow.hPow (β := ℕ) (fundSystem K i) sorry)⁻¹
h_tors : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) ζ
⊢ ∃! ζe : ↥(torsion K) × (Fin (rank K) → ℤ), x = (↑ζe.1 : (𝓞 K)ˣ) * ∏ i : Fin (rank K), fundSystem K i ^ ζe.2 i
have refine_1 : (fun (ζe : ↥(torsion K) × (Fin (rank K) → ℤ)) ↦ x = (↑ζe.1 : (𝓞 K)ˣ) * ∏ i : Fin (rank K), fundSystem K i ^ ζe.2 i)
  (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) ζ h_tors, sorry)
Tactic: refine ⟨⟨⟨ζ, h_tors⟩, ((basisModTorsion K).repr (Additive.ofMul ↑x) : Fin (rank K) → ℤ)⟩, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
x : (𝓞 K)ˣ
ζ : (𝓞 K)ˣ := x * (∏ i : Fin (rank K), HPow.hPow (β := ℕ) (fundSystem K i) sorry)⁻¹
h_tors : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) ζ
refine_1 :
  (fun (ζe : ↥(torsion K) × (Fin (rank K) → ℤ)) ↦ x = (↑ζe.1 : (𝓞 K)ˣ) * ∏ i : Fin (rank K), fundSystem K i ^ ζe.2 i)
    (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) ζ h_tors, sorry)
⊢ ∃! ζe : ↥(torsion K) × (Fin (rank K) → ℤ), x = (↑ζe.1 : (𝓞 K)ˣ) * ∏ i : Fin (rank K), fundSystem K i ^ ζe.2 i
have refine_2 : ∀ (y : ↥(torsion K) × (Fin (rank K) → ℤ)),
  (fun (ζe : ↥(torsion K) × (Fin (rank K) → ℤ)) ↦ x = (↑ζe.1 : (𝓞 K)ˣ) * ∏ i : Fin (rank K), fundSystem K i ^ ζe.2 i)
      y →
    y = (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) ζ h_tors, sorry)
Tactic: refine ⟨⟨⟨ζ, h_tors⟩, ((basisModTorsion K).repr (Additive.ofMul ↑x) : Fin (rank K) → ℤ)⟩, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
x : (𝓞 K)ˣ
ζ : (𝓞 K)ˣ := x * (∏ i : Fin (rank K), HPow.hPow (β := ℕ) (fundSystem K i) sorry)⁻¹
h_tors : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) ζ
⊢ (fun (ζe : ↥(torsion K) × (Fin (rank K) → ℤ)) ↦ x = (↑ζe.1 : (𝓞 K)ˣ) * ∏ i : Fin (rank K), fundSystem K i ^ ζe.2 i)
    (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) ζ h_tors, sorry)
have refine_1 : x =
  x * (∏ i : Fin (rank K), HPow.hPow (β := ℕ) (fundSystem K i) sorry)⁻¹ *
    ∏ x : Fin (rank K), HPow.hPow (β := ℤ) (fundSystem K x) (sorry x)
Tactic: simp only [ζ, _root_.inv_mul_cancel_right]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
x : (𝓞 K)ˣ
ζ : (𝓞 K)ˣ := x * (∏ i : Fin (rank K), HPow.hPow (β := ℕ) (fundSystem K i) sorry)⁻¹
h_tors : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) ζ
η : Fin (rank K) → ℤ
ζ' : (𝓞 K)ˣ
h_tors' : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) ζ'
hf :
  x =
    (↑(Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) ζ' h_tors', η).1 :
        (𝓞 K)ˣ) *
      ∏ i : Fin (rank K),
        fundSystem K i ^
          (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) ζ' h_tors', η).2 i
⊢ (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) ζ' h_tors', η) =
    (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) ζ h_tors, sorry)
have refine_2 : ζ' = x * (∏ i : Fin (rank K), HPow.hPow (β := ℕ) (fundSystem K i) sorry)⁻¹
Tactic: simp only [ζ, ← fun_eq_repr K h_tors' hf, Prod.mk.injEq, Subtype.mk.injEq, and_true]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
x : (𝓞 K)ˣ
ζ : (𝓞 K)ˣ := x * (∏ i : Fin (rank K), HPow.hPow (β := ℕ) (fundSystem K i) sorry)⁻¹
h_tors : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) ζ
η : Fin (rank K) → ℤ
ζ' : (𝓞 K)ˣ
h_tors' : Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) ζ'
hf :
  x =
    (↑(Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) ζ' h_tors', η).1 :
        (𝓞 K)ˣ) *
      ∏ i : Fin (rank K),
        fundSystem K i ^
          (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) ζ' h_tors', η).2 i
⊢ ζ' = x * (∏ i : Fin (rank K), HPow.hPow (β := ℕ) (fundSystem K i) sorry)⁻¹
have refine_2 : ζ' =
  ((↑(Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) ζ' h_tors', η).1 :
        (𝓞 K)ˣ) *
      ∏ i : Fin (rank K),
        fundSystem K i ^
          (Subtype.mk (p := fun (x : (𝓞 K)ˣ) ↦ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K) x) ζ' h_tors', η).2
            i) *
    (∏ i : Fin (rank K), HPow.hPow (β := ℕ) (fundSystem K i) sorry)⁻¹
Tactic: nth_rewrite 1 [hf]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
⊢ Eq (α := Subgroup (𝓞 K)ˣ) (Subgroup.closure (Set.range (ι := Fin (rank K)) (fundSystem K)) ⊔ torsion K) ⊤
have rw : ∀ (x : (𝓞 K)ˣ),
  Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K ⊔ Subgroup.closure (Set.range (ι := Fin (rank K)) (fundSystem K))) x
Tactic: rw [Subgroup.eq_top_iff', sup_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.DirichletTheorem (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
x : (𝓞 K)ˣ
⊢ Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K ⊔ Subgroup.closure (Set.range (ι := Fin (rank K)) (fundSystem K))) x
have obtain : ∀ (c : ↥(torsion K) × (Fin (rank K) → ℤ)),
  (∀ (y : ↥(torsion K) × (Fin (rank K) → ℤ)),
      (fun (ζe : ↥(torsion K) × (Fin (rank K) → ℤ)) ↦
            (↑c.1 : (𝓞 K)ˣ) * ∏ i : Fin (rank K), fundSystem K i ^ c.2 i =
              (↑ζe.1 : (𝓞 K)ˣ) * ∏ i : Fin (rank K), fundSystem K i ^ ζe.2 i)
          y →
        y = c) →
    Membership.mem (γ := Subgroup (𝓞 K)ˣ) (torsion K ⊔ Subgroup.closure (Set.range (ι := Fin (rank K)) (fundSystem K)))
      ((↑c.1 : (𝓞 K)ˣ) * ∏ i : Fin (rank K), fundSystem K i ^ c.2 i)
Tactic: obtain ⟨c, rfl, _⟩ := exist_unique_eq_mul_prod K x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
⊢ Fintype.card (Fin (rank K)) = Fintype.card.{u_1} { w : InfinitePlace K // w ≠ w₀ }
have rw : Fintype.card (Fin (rank K)) =
  Fintype.card.{u_1} (InfinitePlace K) - Fintype.card.{u_1} { x : InfinitePlace K // x = w₀ }
Tactic: rw [Fintype.card_subtype_compl, Fintype.card_ofSubsingleton, Fintype.card_fin, rank]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
rw :
  Fintype.card (Fin (rank K)) =
    Fintype.card.{u_1} (InfinitePlace K) - Fintype.card.{u_1} { x : InfinitePlace K // x = w₀ }
⊢ Fintype.card (Fin (rank K)) = Fintype.card.{u_1} { w : InfinitePlace K // w ≠ w₀ }
have rw₁ : Fintype.card (Fin (rank K)) = Fintype.card.{u_1} (InfinitePlace K) - 1
Tactic: rw [Fintype.card_subtype_compl, Fintype.card_ofSubsingleton, Fintype.card_fin, rank]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
rw :
  Fintype.card (Fin (rank K)) =
    Fintype.card.{u_1} (InfinitePlace K) - Fintype.card.{u_1} { x : InfinitePlace K // x = w₀ }
rw₁ : Fintype.card (Fin (rank K)) = Fintype.card.{u_1} (InfinitePlace K) - 1
⊢ Fintype.card (Fin (rank K)) = Fintype.card.{u_1} { w : InfinitePlace K // w ≠ w₀ }
have rw₂ : rank K = Fintype.card.{u_1} (InfinitePlace K) - 1
Tactic: rw [Fintype.card_subtype_compl, Fintype.card_ofSubsingleton, Fintype.card_fin, rank]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
rw :
  Fintype.card (Fin (rank K)) =
    Fintype.card.{u_1} (InfinitePlace K) - Fintype.card.{u_1} { x : InfinitePlace K // x = w₀ }
rw₁ : Fintype.card (Fin (rank K)) = Fintype.card.{u_1} (InfinitePlace K) - 1
rw₂ : rank K = Fintype.card.{u_1} (InfinitePlace K) - 1
⊢ Fintype.card (Fin (rank K)) = Fintype.card.{u_1} { w : InfinitePlace K // w ≠ w₀ }
have rw₃ : Fintype.card.{u_1} (InfinitePlace K) - 1 = Fintype.card.{u_1} (InfinitePlace K) - 1
Tactic: rw [Fintype.card_subtype_compl, Fintype.card_ofSubsingleton, Fintype.card_fin, rank]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
hu : IsMaxRank u
⊢ Eq (α := AddSubgroup (logSpace K))
    (toAddSubgroup (module_M := AddCommGroup.toIntModule (logSpace K))
      (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hu) : (a : Fin (rank K)) → logSpace K))))
    (AddSubgroup.map (logEmbedding K)
      ((Subgroup.toAddSubgroup : (a : Subgroup (𝓞 K)ˣ) → AddSubgroup (Additive (𝓞 K)ˣ))
        (Subgroup.closure (Set.range u))))
have rw : Eq (α := AddSubgroup (logSpace K))
  (toAddSubgroup (module_M := AddCommGroup.toIntModule (logSpace K))
    (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hu) : (a : Fin (rank K)) → logSpace K))))
  (toAddSubgroup (module_M := AddCommGroup.toIntModule (logSpace K))
    (span ℤ
      (Set.image (α := Additive (𝓞 K)ˣ) (⇑(logEmbedding K) : (a : Additive (𝓞 K)ˣ) → logSpace K)
        ((⇑Additive.toMul : (a : Additive (𝓞 K)ˣ) → (𝓞 K)ˣ) ⁻¹' Set.range u))))
Tactic: rw [Subgroup.toAddSubgroup_closure, AddMonoidHom.map_closure, ← span_int_eq_addSubgroupClosure]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
hu : IsMaxRank u
⊢ Eq (α := AddSubgroup (logSpace K))
    (toAddSubgroup (module_M := AddCommGroup.toIntModule (logSpace K))
      (span ℤ (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hu) : (a : Fin (rank K)) → logSpace K))))
    (toAddSubgroup (module_M := AddCommGroup.toIntModule (logSpace K))
      (span ℤ
        (Set.image (α := Additive (𝓞 K)ˣ) (⇑(logEmbedding K) : (a : Additive (𝓞 K)ˣ) → logSpace K)
          ((⇑Additive.toMul : (a : Additive (𝓞 K)ˣ) → (𝓞 K)ˣ) ⁻¹' Set.range u))))
have e_p : Eq (α := Set (logSpace K)) (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hu) : (a : Fin (rank K)) → logSpace K))
  (Set.image (α := Additive (𝓞 K)ˣ) (⇑(logEmbedding K) : (a : Additive (𝓞 K)ˣ) → logSpace K)
    ((⇑Additive.toMul : (a : Additive (𝓞 K)ˣ) → (𝓞 K)ˣ) ⁻¹' Set.range u))
Tactic: congr
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
u : Fin (rank K) → (𝓞 K)ˣ
hu : IsMaxRank u
⊢ Eq (α := Set (logSpace K)) (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hu) : (a : Fin (rank K)) → logSpace K))
    (Set.image (α := Additive (𝓞 K)ˣ) (⇑(logEmbedding K) : (a : Additive (𝓞 K)ˣ) → logSpace K)
      ((⇑Additive.toMul : (a : Additive (𝓞 K)ˣ) → (𝓞 K)ˣ) ⁻¹' Set.range u))
have e_p : ∀ (x : logSpace K),
  Membership.mem (γ := Set (logSpace K))
      (Set.range (ι := Fin (rank K)) (⇑(basisOfIsMaxRank hu) : (a : Fin (rank K)) → logSpace K)) x ↔
    Membership.mem (γ := Set (logSpace K))
      (Set.image (α := Additive (𝓞 K)ˣ) (⇑(logEmbedding K) : (a : Additive (𝓞 K)ˣ) → logSpace K)
        ((⇑Additive.toMul : (a : Additive (𝓞 K)ˣ) → (𝓞 K)ˣ) ⁻¹' Set.range u))
      x
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
s : Subgroup (𝓞 K)ˣ
⊢ s.FiniteIndex ↔ (s ⊔ torsion K).FiniteIndex
have refine : ∀ (h : (s ⊔ torsion K).FiniteIndex), s.FiniteIndex
Tactic: refine ⟨fun h ↦ Subgroup.finiteIndex_of_le le_sup_left, fun h ↦ ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
s : Subgroup (𝓞 K)ˣ
h : (s ⊔ torsion K).FiniteIndex
⊢ s.FiniteIndex
have rw : s.relIndex (s ⊔ torsion K) * (s ⊔ torsion K).index ≠ 0
Tactic: rw [Subgroup.finiteIndex_iff, ← Subgroup.relIndex_mul_index (le_sup_left : s ≤ s ⊔ torsion K)]
-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.Units.Regulator (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
s : Subgroup (𝓞 K)ˣ
h : (s ⊔ torsion K).FiniteIndex
⊢ s.relIndex (s ⊔ torsion K) * (s ⊔ torsion K).index ≠ 0
have refine : s.relIndex (s ⊔ torsion K) ≠ 0
Tactic: refine Nat.mul_ne_zero ?_ (Subgroup.finiteIndex_iff.mp h)