Tsum lambert

theorem tsum_lambert (q : k) (hq : valuation k q < 1) (c : ℕ → ℤ) :    ∑' m : ℕ, ((c (m + 1) : ℤ) : k) * q ^ (m + 1) / (1 - q ^ (m + 1)) =      ∑' N : ℕ, ((∑ d ∈ N.divisors, c d : ℤ) : k) * q ^ N := by  -- powers of `q` stay in the open unit disc  have hqpow : ∀ n : ℕ+, valuation k (q ^ (n : ℕ)) < 1 := fun n ↦ by    rw [map_pow]    calc valuation k q ^ (n : ℕ) ≤ valuation k q ^ 1 :=          pow_le_pow_right_of_le_one' hq.le n.pos      _ = valuation k q := pow_one _      _ < 1 := hq  -- each row of the double series is a geometric series  have hgeo : ∀ m : ℕ+, HasSum (fun j : ℕ ↦ q ^ ((m : ℕ) * (j + 1)))      (q ^ (m : ℕ) / (1 - q ^ (m : ℕ))) := fun m ↦ by    simpa only [← pow_mul] using hasSum_geometric_succ (hqpow m)  -- the double series is summable, its terms tending to zero nonarchimedeanly  have hsum : Summable fun p : ℕ+ × ℕ+ ↦ ((c p.1 : ℤ) : k) * q ^ ((p.1 : ℕ) * (p.2 : ℕ)) := by    refine summable_of_valuation_le_pow hq (fun p ↦ (p.1 : ℕ) * (p.2 : ℕ)) (fun N ↦ ?_) fun p ↦ ?_    · refine (((Set.finite_Iio N).preimage PNat.coe_injective.injOn).prod        ((Set.finite_Iio N).preimage PNat.coe_injective.injOn)).subset fun p hp ↦ ?_      have h1 : (p.1 : ℕ) ≤ (p.1 : ℕ) * (p.2 : ℕ) := Nat.le_mul_of_pos_right _ p.2.pos      have h2 : (p.2 : ℕ) ≤ (p.1 : ℕ) * (p.2 : ℕ) := Nat.le_mul_of_pos_left _ p.1.pos      exact Set.mem_prod.mpr ⟨Set.mem_preimage.mpr (Set.mem_Iio.mpr (lt_of_le_of_lt h1 hp)),        Set.mem_preimage.mpr (Set.mem_Iio.mpr (lt_of_le_of_lt h2 hp))⟩    · rw [map_mul, map_pow]      simpa using mul_le_mul_left (valuation_intCast_le_one _)        (valuation k q ^ ((p.1 : ℕ) * (p.2 : ℕ)))  calc ∑' m : ℕ, ((c (m + 1) : ℤ) : k) * q ^ (m + 1) / (1 - q ^ (m + 1))      = ∑' m : ℕ+, ((c m : ℤ) : k) * q ^ (m : ℕ) / (1 - q ^ (m : ℕ)) :=        (tsum_pnat_eq_tsum_succ (f := fun n ↦ ((c n : ℤ) : k) * q ^ n / (1 - q ^ n))).symm    _ = ∑' N : ℕ+, (∑ d ∈ (N : ℕ).divisors, ((c d : ℤ) : k)) * q ^ (N : ℕ) :=        tsum_lambert_of_summable q (fun d ↦ ((c d : ℤ) : k)) hgeo hsum    _ = ∑' N : ℕ+, ((∑ d ∈ (N : ℕ).divisors, c d : ℤ) : k) * q ^ (N : ℕ) :=        tsum_congr fun N ↦ by push_cast; ring    _ = ∑' N : ℕ, ((∑ d ∈ N.divisors, c d : ℤ) : k) * q ^ N := by        refine PNat.coe_injective.tsum_eq          (f := fun N : ℕ ↦ ((∑ d ∈ N.divisors, c d : ℤ) : k) * q ^ N) fun x hx ↦ ?_        cases x with        | zero => simp at hx        | succ n => exact ⟨n.succPNat, rfl⟩