-- Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean
theorem exist_unique_eq_mul_prod (x : (𝓞 K)ˣ) : ∃! ζe : torsion K × (Fin (rank K) → ℤ),
    x = ζe.1 * ∏ i, (fundSystem K i) ^ (ζe.2 i)

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 ^ sorry)⁻¹
h_tors : ζ ∈ torsion K
⊢ ∃! ζe, x = ↑ζe.1 * ∏ i : Fin (rank K), fundSystem K i ^ ζe.2 i

Tactic:
refine ⟨⟨⟨ζ, h_tors⟩, ((basisModTorsion K).repr (Additive.ofMul ↑x) : Fin (rank K) → ℤ)⟩, ?_, ?_⟩

Result:
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 ^ sorry)⁻¹
h_tors : ζ ∈ torsion K
⊢ (fun ζe => x = ↑ζe.1 * ∏ i : Fin (rank K), fundSystem K i ^ ζe.2 i) (⟨ζ, h_tors⟩, sorry)

case refine_2
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 ^ sorry)⁻¹
h_tors : ζ ∈ torsion K
⊢ ∀ (y : ↥(torsion K) × (Fin (rank K) → ℤ)),
    (fun ζe => x = ↑ζe.1 * ∏ i : Fin (rank K), fundSystem K i ^ ζe.2 i) y → y = (⟨ζ, h_tors⟩, sorry)

Proof state:
case refine_2.mk.mk
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 ^ sorry)⁻¹
h_tors : ζ ∈ torsion K
η : Fin (rank K) → ℤ
ζ' : (𝓞 K)ˣ
h_tors' : ζ' ∈ torsion K
hf : x = ↑(⟨ζ', h_tors'⟩, η).1 * ∏ i : Fin (rank K), fundSystem K i ^ (⟨ζ', h_tors'⟩, η).2 i
⊢ ζ' = x * (∏ i : Fin (rank K), fundSystem K i ^ sorry)⁻¹

Tactic:
nth_rewrite 1 [hf]

Result:
case refine_2.mk.mk
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 ^ sorry)⁻¹
h_tors : ζ ∈ torsion K
η : Fin (rank K) → ℤ
ζ' : (𝓞 K)ˣ
h_tors' : ζ' ∈ torsion K
hf : x = ↑(⟨ζ', h_tors'⟩, η).1 * ∏ i : Fin (rank K), fundSystem K i ^ (⟨ζ', h_tors'⟩, η).2 i
⊢ ζ' =
    (↑(⟨ζ', h_tors'⟩, η).1 * ∏ i : Fin (rank K), fundSystem K i ^ (⟨ζ', h_tors'⟩, η).2 i) *
      (∏ i : Fin (rank K), fundSystem K i ^ sorry)⁻¹

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (𝓞 K)ˣ
⊢ ∃! ζe, x = ↑ζe.1 * ∏ i : Fin (rank K), fundSystem K i ^ ζe.2 i

Tactic:
let ζ := x * (∏ i, (fundSystem K i) ^ ((basisModTorsion K).repr (Additive.ofMul ↑x) i))⁻¹

Result:
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 ^ sorry)⁻¹
⊢ ∃! ζe, x = ↑ζe.1 * ∏ i : Fin (rank K), fundSystem K i ^ ζe.2 i

Proof state:
case refine_2.mk.mk
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 ^ sorry)⁻¹
h_tors : ζ ∈ torsion K
η : Fin (rank K) → ℤ
ζ' : (𝓞 K)ˣ
h_tors' : ζ' ∈ torsion K
hf : x = ↑(⟨ζ', h_tors'⟩, η).1 * ∏ i : Fin (rank K), fundSystem K i ^ (⟨ζ', h_tors'⟩, η).2 i
⊢ (⟨ζ', h_tors'⟩, η) = (⟨ζ, h_tors⟩, sorry)

Tactic:
simp only [ζ, ← fun_eq_repr K h_tors' hf, Prod.mk.injEq, Subtype.mk.injEq, and_true]

Result:
case refine_2.mk.mk
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 ^ sorry)⁻¹
h_tors : ζ ∈ torsion K
η : Fin (rank K) → ℤ
ζ' : (𝓞 K)ˣ
h_tors' : ζ' ∈ torsion K
hf : x = ↑(⟨ζ', h_tors'⟩, η).1 * ∏ i : Fin (rank K), fundSystem K i ^ (⟨ζ', h_tors'⟩, η).2 i
⊢ ζ' = x * (∏ i : Fin (rank K), fundSystem K i ^ sorry)⁻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 ^ sorry)⁻¹
⊢ ∃! ζe, x = ↑ζe.1 * ∏ i : Fin (rank K), fundSystem K i ^ ζe.2 i

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

Result:
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 ^ sorry)⁻¹
h_tors : ζ ∈ torsion K
⊢ ∃! ζe, x = ↑ζe.1 * ∏ i : Fin (rank K), fundSystem K i ^ ζe.2 i

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 ^ sorry)⁻¹
⊢ ζ ∈ torsion K

Tactic:
rw [← QuotientGroup.eq_one_iff, QuotientGroup.mk_mul, QuotientGroup.mk_inv, ← ofMul_eq_zero,
  ofMul_mul, ofMul_inv, QuotientGroup.mk_prod, ofMul_prod]

Result:
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 ^ sorry)⁻¹
⊢ Additive.ofMul ↑x + -∑ i : Fin (rank K), Additive.ofMul ↑(fundSystem K i ^ sorry) = 0

Proof state:
case refine_2
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 ^ sorry)⁻¹
h_tors : ζ ∈ torsion K
⊢ ∀ (y : ↥(torsion K) × (Fin (rank K) → ℤ)),
    (fun ζe => x = ↑ζe.1 * ∏ i : Fin (rank K), fundSystem K i ^ ζe.2 i) y → y = (⟨ζ, h_tors⟩, sorry)

Tactic:
rintro ⟨⟨ζ', h_tors'⟩, η⟩ hf

Result:
case refine_2.mk.mk
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 ^ sorry)⁻¹
h_tors : ζ ∈ torsion K
η : Fin (rank K) → ℤ
ζ' : (𝓞 K)ˣ
h_tors' : ζ' ∈ torsion K
hf : x = ↑(⟨ζ', h_tors'⟩, η).1 * ∏ i : Fin (rank K), fundSystem K i ^ (⟨ζ', h_tors'⟩, η).2 i
⊢ (⟨ζ', h_tors'⟩, η) = (⟨ζ, h_tors⟩, sorry)

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 ^ sorry)⁻¹
h_tors : ζ ∈ torsion K
⊢ (fun ζe => x = ↑ζe.1 * ∏ i : Fin (rank K), fundSystem K i ^ ζe.2 i) (⟨ζ, h_tors⟩, sorry)

Tactic:
simp only [ζ, _root_.inv_mul_cancel_right]

Result:
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 ^ sorry)⁻¹
h_tors : ζ ∈ torsion K
⊢ x = x * (∏ i : Fin (rank K), fundSystem K i ^ sorry)⁻¹ * ∏ x : Fin (rank K), fundSystem K x ^ sorry x