l3lab/ntp-mathlib huggingface dataset:
set_option synthInstance.maxHeartbeats 27000 in
/-- The linear equivalence between `unitLattice` and `(𝓞 K)ˣ ⧸ (torsion K)` as an additive
`ℤ`-module. -/
def unitLatticeEquiv : (unitLattice K) ≃ₗ[ℤ] Additive ((𝓞 K)ˣ ⧸ (torsion K)) := by
  refine AddEquiv.toIntLinearEquiv ?_
  rw [unitLattice, ← AddMonoidHom.range_eq_map (logEmbedding K)]
  refine (QuotientAddGroup.quotientKerEquivRange (logEmbedding K)).symm.trans ?_
  refine (QuotientAddGroup.quotientAddEquivOfEq ?_).trans
    (QuotientAddGroup.quotientKerEquivOfSurjective
      (MonoidHom.toAdditive (QuotientGroup.mk' (torsion K))) (fun x => ?_))
  · ext
    rw [MonoidHom.coe_toAdditive_ker, QuotientGroup.ker_mk', AddMonoidHom.mem_ker,
      logEmbedding_eq_zero_iff]
    rfl
  · refine ⟨Additive.ofMul x.out', ?_⟩
    

Proof state:
case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : Additive ((↥(𝓞 K))ˣ ⧸ torsion K)
⊢ (MonoidHom.toAdditive (QuotientGroup.mk' (torsion K))) (Additive.ofMul (Quotient.out' x)) = x

Next tactic:
simp only [MonoidHom.toAdditive_apply_apply, toMul_ofMul, QuotientGroup.mk'_apply,
      QuotientGroup.out_eq']
set_option synthInstance.maxHeartbeats 27000 in
/-- The linear equivalence between `unitLattice` and `(𝓞 K)ˣ ⧸ (torsion K)` as an additive
`ℤ`-module. -/
def unitLatticeEquiv : (unitLattice K) ≃ₗ[ℤ] Additive ((𝓞 K)ˣ ⧸ (torsion K)) := by
  refine AddEquiv.toIntLinearEquiv ?_
  rw [unitLattice, ← AddMonoidHom.range_eq_map (logEmbedding K)]
  refine (QuotientAddGroup.quotientKerEquivRange (logEmbedding K)).symm.trans ?_
  refine (QuotientAddGroup.quotientAddEquivOfEq ?_).trans
    (QuotientAddGroup.quotientKerEquivOfSurjective
      (MonoidHom.toAdditive (QuotientGroup.mk' (torsion K))) (fun x => ?_))
  · ext
    rw [MonoidHom.coe_toAdditive_ker, QuotientGroup.ker_mk', AddMonoidHom.mem_ker,
      logEmbedding_eq_zero_iff]
    rfl
  · refine ⟨Additive.ofMul x.out', ?_⟩
    simp only [MonoidHom.toAdditive_apply_apply, toMul_ofMul, QuotientGroup.mk'_apply,
      QuotientGroup.out_eq']
    

Proof state:
case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : Additive ((↥(𝓞 K))ˣ ⧸ torsion K)
⊢ Additive.ofMul x = x

Next tactic:
rfl
instance : Module.Finite ℤ (Additive (𝓞 K)ˣ) := by
  

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ Module.Finite ℤ (Additive (↥(𝓞 K))ˣ)

Next tactic:
rw [Module.finite_def]
instance : Module.Finite ℤ (Additive (𝓞 K)ˣ) := by
  rw [Module.finite_def]
  

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ Submodule.FG ⊤

Next tactic:
refine Submodule.fg_of_fg_map_of_fg_inf_ker
    (MonoidHom.toAdditive (QuotientGroup.mk' (torsion K))).toIntLinearMap ?_ ?_
instance : Module.Finite ℤ (Additive (𝓞 K)ˣ) := by
  rw [Module.finite_def]
  refine Submodule.fg_of_fg_map_of_fg_inf_ker
    (MonoidHom.toAdditive (QuotientGroup.mk' (torsion K))).toIntLinearMap ?_ ?_
  · 

Proof state:
case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ Submodule.FG (Submodule.map (AddMonoidHom.toIntLinearMap (MonoidHom.toAdditive (QuotientGroup.mk' (torsion K)))) ⊤)

Next tactic:
rw [Submodule.map_top, LinearMap.range_eq_top.mpr
      (by exact QuotientGroup.mk'_surjective (torsion K)), ← Module.finite_def]
instance : Module.Finite ℤ (Additive (𝓞 K)ˣ) := by
  rw [Module.finite_def]
  refine Submodule.fg_of_fg_map_of_fg_inf_ker
    (MonoidHom.toAdditive (QuotientGroup.mk' (torsion K))).toIntLinearMap ?_ ?_
  · rw [Submodule.map_top, LinearMap.range_eq_top.mpr
      (by 

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ Function.Surjective ⇑(AddMonoidHom.toIntLinearMap (MonoidHom.toAdditive (QuotientGroup.mk' (torsion K))))

Next tactic:
exact QuotientGroup.mk'_surjective (torsion K)
instance : Module.Finite ℤ (Additive (𝓞 K)ˣ) := by
  rw [Module.finite_def]
  refine Submodule.fg_of_fg_map_of_fg_inf_ker
    (MonoidHom.toAdditive (QuotientGroup.mk' (torsion K))).toIntLinearMap ?_ ?_
  · rw [Submodule.map_top, LinearMap.range_eq_top.mpr
      (by exact QuotientGroup.mk'_surjective (torsion K)), ← Module.finite_def]
    

Proof state:
case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ Module.Finite ℤ (Additive ((↥(𝓞 K))ˣ ⧸ torsion K))

Next tactic:
infer_instance
instance : Module.Finite ℤ (Additive (𝓞 K)ˣ) := by
  rw [Module.finite_def]
  refine Submodule.fg_of_fg_map_of_fg_inf_ker
    (MonoidHom.toAdditive (QuotientGroup.mk' (torsion K))).toIntLinearMap ?_ ?_
  · rw [Submodule.map_top, LinearMap.range_eq_top.mpr
      (by exact QuotientGroup.mk'_surjective (torsion K)), ← Module.finite_def]
    infer_instance
  · 

Proof state:
case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ Submodule.FG (⊤ ⊓ LinearMap.ker (AddMonoidHom.toIntLinearMap (MonoidHom.toAdditive (QuotientGroup.mk' (torsion K)))))

Next tactic:
rw [inf_of_le_right le_top, AddMonoidHom.coe_toIntLinearMap_ker, MonoidHom.coe_toAdditive_ker,
      QuotientGroup.ker_mk', Submodule.fg_iff_add_subgroup_fg,
      AddSubgroup.toIntSubmodule_toAddSubgroup, ← AddGroup.fg_iff_addSubgroup_fg]
instance : Module.Finite ℤ (Additive (𝓞 K)ˣ) := by
  rw [Module.finite_def]
  refine Submodule.fg_of_fg_map_of_fg_inf_ker
    (MonoidHom.toAdditive (QuotientGroup.mk' (torsion K))).toIntLinearMap ?_ ?_
  · rw [Submodule.map_top, LinearMap.range_eq_top.mpr
      (by exact QuotientGroup.mk'_surjective (torsion K)), ← Module.finite_def]
    infer_instance
  · rw [inf_of_le_right le_top, AddMonoidHom.coe_toIntLinearMap_ker, MonoidHom.coe_toAdditive_ker,
      QuotientGroup.ker_mk', Submodule.fg_iff_add_subgroup_fg,
      AddSubgroup.toIntSubmodule_toAddSubgroup, ← AddGroup.fg_iff_addSubgroup_fg]
    

Proof state:
case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ AddGroup.FG ↥(Subgroup.toAddSubgroup (torsion K))

Next tactic:
have : Finite (Subgroup.toAddSubgroup (torsion K)) := (inferInstance : Finite (torsion K))
instance : Module.Finite ℤ (Additive (𝓞 K)ˣ) := by
  rw [Module.finite_def]
  refine Submodule.fg_of_fg_map_of_fg_inf_ker
    (MonoidHom.toAdditive (QuotientGroup.mk' (torsion K))).toIntLinearMap ?_ ?_
  · rw [Submodule.map_top, LinearMap.range_eq_top.mpr
      (by exact QuotientGroup.mk'_surjective (torsion K)), ← Module.finite_def]
    infer_instance
  · rw [inf_of_le_right le_top, AddMonoidHom.coe_toIntLinearMap_ker, MonoidHom.coe_toAdditive_ker,
      QuotientGroup.ker_mk', Submodule.fg_iff_add_subgroup_fg,
      AddSubgroup.toIntSubmodule_toAddSubgroup, ← AddGroup.fg_iff_addSubgroup_fg]
    have : Finite (Subgroup.toAddSubgroup (torsion K)) := (inferInstance : Finite (torsion K))
    

Proof state:
case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
this : Finite ↥(Subgroup.toAddSubgroup (torsion K))
⊢ AddGroup.FG ↥(Subgroup.toAddSubgroup (torsion K))

Next tactic:
exact AddGroup.fg_of_finite
instance : Monoid.FG (𝓞 K)ˣ := by
  

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ Monoid.FG (↥(𝓞 K))ˣ

Next tactic:
rw [Monoid.fg_iff_add_fg, ← AddGroup.fg_iff_addMonoid_fg, ← Module.Finite.iff_addGroup_fg]
instance : Monoid.FG (𝓞 K)ˣ := by
  rw [Monoid.fg_iff_add_fg, ← AddGroup.fg_iff_addMonoid_fg, ← Module.Finite.iff_addGroup_fg]
  

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ Module.Finite ℤ (Additive (↥(𝓞 K))ˣ)

Next tactic:
infer_instance
theorem rank_modTorsion :
    FiniteDimensional.finrank ℤ (Additive ((𝓞 K)ˣ ⧸ (torsion K))) = rank K := by
  

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ finrank ℤ (Additive ((↥(𝓞 K))ˣ ⧸ torsion K)) = rank K

Next tactic:
rw [← LinearEquiv.finrank_eq (unitLatticeEquiv K), unitLattice_rank]
/-- A basis of the quotient `(𝓞 K)ˣ ⧸ (torsion K)` seen as an additive ℤ-module. -/
def basisModTorsion : Basis (Fin (rank K)) ℤ (Additive ((𝓞 K)ˣ ⧸ (torsion K))) :=
  Basis.reindex (Module.Free.chooseBasis ℤ _) (Fintype.equivOfCardEq <| by
    

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ Fintype.card (Module.Free.ChooseBasisIndex ℤ (Additive ((↥(𝓞 K))ˣ ⧸ torsion K))) = Fintype.card (Fin (rank K))

Next tactic:
rw [← FiniteDimensional.finrank_eq_card_chooseBasisIndex, rank_modTorsion, Fintype.card_fin]
/-- The exponents that appear in the unique decomposition of a unit as the product of
a root of unity and powers of the units of the fundamental system `fundSystem` (see
`exist_unique_eq_mul_prod`) are given by the representation of the unit on `basisModTorsion`. -/
theorem fun_eq_repr {x ζ : (𝓞 K)ˣ} {f : Fin (rank K) → ℤ} (hζ : ζ ∈ torsion K)
    (h : x = ζ * ∏ i, (fundSystem K i) ^ (f i)) :
    f = (basisModTorsion K).repr (Additive.ofMul ↑x) := by
  

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x ζ : (↥(𝓞 K))ˣ
f : Fin (rank K) → ℤ
hζ : ζ ∈ torsion K
h : x = ζ * ∏ i : Fin (rank K), fundSystem K i ^ f i
⊢ f = ⇑((basisModTorsion K).repr (Additive.ofMul ↑x))

Next tactic:
suffices Additive.ofMul ↑x = ∑ i, (f i) • (basisModTorsion K i) by
    rw [← (basisModTorsion K).repr_sum_self f, ← this]
/-- The exponents that appear in the unique decomposition of a unit as the product of
a root of unity and powers of the units of the fundamental system `fundSystem` (see
`exist_unique_eq_mul_prod`) are given by the representation of the unit on `basisModTorsion`. -/
theorem fun_eq_repr {x ζ : (𝓞 K)ˣ} {f : Fin (rank K) → ℤ} (hζ : ζ ∈ torsion K)
    (h : x = ζ * ∏ i, (fundSystem K i) ^ (f i)) :
    f = (basisModTorsion K).repr (Additive.ofMul ↑x) := by
  suffices Additive.ofMul ↑x = ∑ i, (f i) • (basisModTorsion K i) by
    

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x ζ : (↥(𝓞 K))ˣ
f : Fin (rank K) → ℤ
hζ : ζ ∈ torsion K
h : x = ζ * ∏ i : Fin (rank K), fundSystem K i ^ f i
this : Additive.ofMul ↑x = ∑ i : Fin (rank K), f i • (basisModTorsion K) i
⊢ f = ⇑((basisModTorsion K).repr (Additive.ofMul ↑x))

Next tactic:
rw [← (basisModTorsion K).repr_sum_self f, ← this]
/-- The exponents that appear in the unique decomposition of a unit as the product of
a root of unity and powers of the units of the fundamental system `fundSystem` (see
`exist_unique_eq_mul_prod`) are given by the representation of the unit on `basisModTorsion`. -/
theorem fun_eq_repr {x ζ : (𝓞 K)ˣ} {f : Fin (rank K) → ℤ} (hζ : ζ ∈ torsion K)
    (h : x = ζ * ∏ i, (fundSystem K i) ^ (f i)) :
    f = (basisModTorsion K).repr (Additive.ofMul ↑x) := by
  suffices Additive.ofMul ↑x = ∑ i, (f i) • (basisModTorsion K i) by
    rw [← (basisModTorsion K).repr_sum_self f, ← this]
  

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x ζ : (↥(𝓞 K))ˣ
f : Fin (rank K) → ℤ
hζ : ζ ∈ torsion K
h : x = ζ * ∏ i : Fin (rank K), fundSystem K i ^ f i
⊢ Additive.ofMul ↑x = ∑ i : Fin (rank K), f i • (basisModTorsion K) i

Next 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]
/-- The exponents that appear in the unique decomposition of a unit as the product of
a root of unity and powers of the units of the fundamental system `fundSystem` (see
`exist_unique_eq_mul_prod`) are given by the representation of the unit on `basisModTorsion`. -/
theorem fun_eq_repr {x ζ : (𝓞 K)ˣ} {f : Fin (rank K) → ℤ} (hζ : ζ ∈ torsion K)
    (h : x = ζ * ∏ i, (fundSystem K i) ^ (f i)) :
    f = (basisModTorsion K).repr (Additive.ofMul ↑x) := by
  suffices Additive.ofMul ↑x = ∑ i, (f i) • (basisModTorsion K i) by
    rw [← (basisModTorsion K).repr_sum_self f, ← this]
  calc
    Additive.ofMul ↑x
    _ = ∑ i, (f i) • Additive.ofMul ↑(fundSystem K i) := by
          

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x ζ : (↥(𝓞 K))ˣ
f : Fin (rank K) → ℤ
hζ : ζ ∈ torsion K
h : x = ζ * ∏ i : Fin (rank K), fundSystem K i ^ f i
⊢ Additive.ofMul ↑x = ∑ i : Fin (rank K), f i • Additive.ofMul ↑(fundSystem K i)

Next tactic:
rw [h, QuotientGroup.mk_mul, (QuotientGroup.eq_one_iff _).mpr hζ, one_mul,
            QuotientGroup.mk_prod, ofMul_prod]
/-- The exponents that appear in the unique decomposition of a unit as the product of
a root of unity and powers of the units of the fundamental system `fundSystem` (see
`exist_unique_eq_mul_prod`) are given by the representation of the unit on `basisModTorsion`. -/
theorem fun_eq_repr {x ζ : (𝓞 K)ˣ} {f : Fin (rank K) → ℤ} (hζ : ζ ∈ torsion K)
    (h : x = ζ * ∏ i, (fundSystem K i) ^ (f i)) :
    f = (basisModTorsion K).repr (Additive.ofMul ↑x) := by
  suffices Additive.ofMul ↑x = ∑ i, (f i) • (basisModTorsion K i) by
    rw [← (basisModTorsion K).repr_sum_self f, ← this]
  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]; 

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x ζ : (↥(𝓞 K))ˣ
f : Fin (rank K) → ℤ
hζ : ζ ∈ torsion K
h : x = ζ * ∏ i : Fin (rank K), fundSystem K i ^ f i
⊢ ∑ i : Fin (rank K), Additive.ofMul ↑(fundSystem K i ^ f i) =
    ∑ i : Fin (rank K), f i • Additive.ofMul ↑(fundSystem K i)

Next tactic:
rfl
/-- The exponents that appear in the unique decomposition of a unit as the product of
a root of unity and powers of the units of the fundamental system `fundSystem` (see
`exist_unique_eq_mul_prod`) are given by the representation of the unit on `basisModTorsion`. -/
theorem fun_eq_repr {x ζ : (𝓞 K)ˣ} {f : Fin (rank K) → ℤ} (hζ : ζ ∈ torsion K)
    (h : x = ζ * ∏ i, (fundSystem K i) ^ (f i)) :
    f = (basisModTorsion K).repr (Additive.ofMul ↑x) := by
  suffices Additive.ofMul ↑x = ∑ i, (f i) • (basisModTorsion K i) by
    rw [← (basisModTorsion K).repr_sum_self f, ← this]
  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
          

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x ζ : (↥(𝓞 K))ˣ
f : Fin (rank K) → ℤ
hζ : ζ ∈ torsion K
h : x = ζ * ∏ i : Fin (rank K), fundSystem K i ^ f i
⊢ ∑ i : Fin (rank K), f i • Additive.ofMul ↑(fundSystem K i) = ∑ i : Fin (rank K), f i • (basisModTorsion K) i

Next tactic:
simp_rw [fundSystem, QuotientGroup.out_eq', ofMul_toMul]
/-- **Dirichlet Unit Theorem**. Any unit `x` of `𝓞 K` can be written uniquely as the product of
a root of unity and powers of the units of the fundamental system `fundSystem`. -/
theorem exist_unique_eq_mul_prod (x : (𝓞 K)ˣ) : ∃! (ζ : torsion K) (e : Fin (rank K) → ℤ),
    x = ζ * ∏ i, (fundSystem K i) ^ (e i) := by
  

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (↥(𝓞 K))ˣ
⊢ ∃! ζ e, x = ↑ζ * ∏ i : Fin (rank K), fundSystem K i ^ e i

Next tactic:
let ζ := x * (∏ i, (fundSystem K i) ^ ((basisModTorsion K).repr (Additive.ofMul ↑x) i))⁻¹
/-- **Dirichlet Unit Theorem**. Any unit `x` of `𝓞 K` can be written uniquely as the product of
a root of unity and powers of the units of the fundamental system `fundSystem`. -/
theorem exist_unique_eq_mul_prod (x : (𝓞 K)ˣ) : ∃! (ζ : torsion K) (e : Fin (rank K) → ℤ),
    x = ζ * ∏ i, (fundSystem K i) ^ (e i) := by
  let ζ := x * (∏ i, (fundSystem K i) ^ ((basisModTorsion K).repr (Additive.ofMul ↑x) i))⁻¹
  

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (↥(𝓞 K))ˣ
ζ : (↥(𝓞 K))ˣ := x * (∏ i : Fin (rank K), fundSystem K i ^ ((basisModTorsion K).repr (Additive.ofMul ↑x)) i)⁻¹
⊢ ∃! ζ e, x = ↑ζ * ∏ i : Fin (rank K), fundSystem K i ^ e i

Next 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
/-- **Dirichlet Unit Theorem**. Any unit `x` of `𝓞 K` can be written uniquely as the product of
a root of unity and powers of the units of the fundamental system `fundSystem`. -/
theorem exist_unique_eq_mul_prod (x : (𝓞 K)ˣ) : ∃! (ζ : torsion K) (e : Fin (rank K) → ℤ),
    x = ζ * ∏ i, (fundSystem K i) ^ (e i) := by
  let ζ := x * (∏ i, (fundSystem K i) ^ ((basisModTorsion K).repr (Additive.ofMul ↑x) i))⁻¹
  have h_tors : ζ ∈ torsion K := by
    

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (↥(𝓞 K))ˣ
ζ : (↥(𝓞 K))ˣ := x * (∏ i : Fin (rank K), fundSystem K i ^ ((basisModTorsion K).repr (Additive.ofMul ↑x)) i)⁻¹
⊢ ζ ∈ torsion K

Next tactic:
rw [← QuotientGroup.eq_one_iff, QuotientGroup.mk_mul, QuotientGroup.mk_inv, ← ofMul_eq_zero,
      ofMul_mul, ofMul_inv, QuotientGroup.mk_prod, ofMul_prod]
/-- **Dirichlet Unit Theorem**. Any unit `x` of `𝓞 K` can be written uniquely as the product of
a root of unity and powers of the units of the fundamental system `fundSystem`. -/
theorem exist_unique_eq_mul_prod (x : (𝓞 K)ˣ) : ∃! (ζ : torsion K) (e : Fin (rank K) → ℤ),
    x = ζ * ∏ i, (fundSystem K i) ^ (e i) := by
  let ζ := x * (∏ i, (fundSystem K i) ^ ((basisModTorsion K).repr (Additive.ofMul ↑x) i))⁻¹
  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]
    

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (↥(𝓞 K))ˣ
ζ : (↥(𝓞 K))ˣ := x * (∏ i : Fin (rank K), fundSystem K i ^ ((basisModTorsion K).repr (Additive.ofMul ↑x)) i)⁻¹
⊢ Additive.ofMul ↑x +
      -∑ i : Fin (rank K), Additive.ofMul ↑(fundSystem K i ^ ((basisModTorsion K).repr (Additive.ofMul ↑x)) i) =
    0

Next tactic:
simp_rw [QuotientGroup.mk_zpow, ofMul_zpow, fundSystem, QuotientGroup.out_eq']
/-- **Dirichlet Unit Theorem**. Any unit `x` of `𝓞 K` can be written uniquely as the product of
a root of unity and powers of the units of the fundamental system `fundSystem`. -/
theorem exist_unique_eq_mul_prod (x : (𝓞 K)ˣ) : ∃! (ζ : torsion K) (e : Fin (rank K) → ℤ),
    x = ζ * ∏ i, (fundSystem K i) ^ (e i) := by
  let ζ := x * (∏ i, (fundSystem K i) ^ ((basisModTorsion K).repr (Additive.ofMul ↑x) i))⁻¹
  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']
    

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (↥(𝓞 K))ˣ
ζ : (↥(𝓞 K))ˣ := x * (∏ i : Fin (rank K), fundSystem K i ^ ((basisModTorsion K).repr (Additive.ofMul ↑x)) i)⁻¹
⊢ Additive.ofMul ↑x +
      -∑ x_1 : Fin (rank K),
          ((basisModTorsion K).repr (Additive.ofMul ↑x)) x_1 •
            Additive.ofMul (Additive.toMul ((basisModTorsion K) x_1)) =
    0

Next tactic:
rw [add_eq_zero_iff_eq_neg, neg_neg]
/-- **Dirichlet Unit Theorem**. Any unit `x` of `𝓞 K` can be written uniquely as the product of
a root of unity and powers of the units of the fundamental system `fundSystem`. -/
theorem exist_unique_eq_mul_prod (x : (𝓞 K)ˣ) : ∃! (ζ : torsion K) (e : Fin (rank K) → ℤ),
    x = ζ * ∏ i, (fundSystem K i) ^ (e i) := by
  let ζ := x * (∏ i, (fundSystem K i) ^ ((basisModTorsion K).repr (Additive.ofMul ↑x) i))⁻¹
  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]
    

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (↥(𝓞 K))ˣ
ζ : (↥(𝓞 K))ˣ := x * (∏ i : Fin (rank K), fundSystem K i ^ ((basisModTorsion K).repr (Additive.ofMul ↑x)) i)⁻¹
⊢ Additive.ofMul ↑x =
    ∑ x_1 : Fin (rank K),
      ((basisModTorsion K).repr (Additive.ofMul ↑x)) x_1 • Additive.ofMul (Additive.toMul ((basisModTorsion K) x_1))

Next tactic:
exact ((basisModTorsion K).sum_repr (Additive.ofMul ↑x)).symm
/-- **Dirichlet Unit Theorem**. Any unit `x` of `𝓞 K` can be written uniquely as the product of
a root of unity and powers of the units of the fundamental system `fundSystem`. -/
theorem exist_unique_eq_mul_prod (x : (𝓞 K)ˣ) : ∃! (ζ : torsion K) (e : Fin (rank K) → ℤ),
    x = ζ * ∏ i, (fundSystem K i) ^ (e i) := by
  let ζ := x * (∏ i, (fundSystem K i) ^ ((basisModTorsion K).repr (Additive.ofMul ↑x) i))⁻¹
  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
  

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (↥(𝓞 K))ˣ
ζ : (↥(𝓞 K))ˣ := x * (∏ i : Fin (rank K), fundSystem K i ^ ((basisModTorsion K).repr (Additive.ofMul ↑x)) i)⁻¹
h_tors : ζ ∈ torsion K
⊢ ∃! ζ e, x = ↑ζ * ∏ i : Fin (rank K), fundSystem K i ^ e i

Next tactic:
refine ⟨⟨ζ, h_tors⟩, ?_, ?_⟩
/-- **Dirichlet Unit Theorem**. Any unit `x` of `𝓞 K` can be written uniquely as the product of
a root of unity and powers of the units of the fundamental system `fundSystem`. -/
theorem exist_unique_eq_mul_prod (x : (𝓞 K)ˣ) : ∃! (ζ : torsion K) (e : Fin (rank K) → ℤ),
    x = ζ * ∏ i, (fundSystem K i) ^ (e i) := by
  let ζ := x * (∏ i, (fundSystem K i) ^ ((basisModTorsion K).repr (Additive.ofMul ↑x) i))⁻¹
  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
  refine ⟨⟨ζ, h_tors⟩, ?_, ?_⟩
  · 

Proof state:
case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (↥(𝓞 K))ˣ
ζ : (↥(𝓞 K))ˣ := x * (∏ i : Fin (rank K), fundSystem K i ^ ((basisModTorsion K).repr (Additive.ofMul ↑x)) i)⁻¹
h_tors : ζ ∈ torsion K
⊢ (fun ζ => ∃! e, x = ↑ζ * ∏ i : Fin (rank K), fundSystem K i ^ e i) { val := ζ, property := h_tors }

Next tactic:
refine ⟨((basisModTorsion K).repr (Additive.ofMul ↑x) : Fin (rank K) → ℤ), ?_, ?_⟩
/-- **Dirichlet Unit Theorem**. Any unit `x` of `𝓞 K` can be written uniquely as the product of
a root of unity and powers of the units of the fundamental system `fundSystem`. -/
theorem exist_unique_eq_mul_prod (x : (𝓞 K)ˣ) : ∃! (ζ : torsion K) (e : Fin (rank K) → ℤ),
    x = ζ * ∏ i, (fundSystem K i) ^ (e i) := by
  let ζ := x * (∏ i, (fundSystem K i) ^ ((basisModTorsion K).repr (Additive.ofMul ↑x) i))⁻¹
  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
  refine ⟨⟨ζ, h_tors⟩, ?_, ?_⟩
  · refine ⟨((basisModTorsion K).repr (Additive.ofMul ↑x) : Fin (rank K) → ℤ), ?_, ?_⟩
    · 

Proof state:
case refine_1.refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (↥(𝓞 K))ˣ
ζ : (↥(𝓞 K))ˣ := x * (∏ i : Fin (rank K), fundSystem K i ^ ((basisModTorsion K).repr (Additive.ofMul ↑x)) i)⁻¹
h_tors : ζ ∈ torsion K
⊢ (fun e => x = ↑{ val := ζ, property := h_tors } * ∏ i : Fin (rank K), fundSystem K i ^ e i)
    ⇑((basisModTorsion K).repr (Additive.ofMul ↑x))

Next tactic:
simp only [_root_.inv_mul_cancel_right]