theorem exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr :
    ∃ (a : 𝓞 K), a ≠ 0 ∧
      |Algebra.norm ℚ (a:K)| ≤ (4 / π) ^ NrComplexPlaces K *
        (finrank ℚ K).factorial / (finrank ℚ K) ^ (finrank ℚ K) * Real.sqrt |discr K| := by
  -- The smallest possible value for `exists_ne_zero_mem_ringOfIntegers_of_norm_le`
  let B := (minkowskiBound K * (convexBodySumFactor K)⁻¹).toReal ^ (1 / (finrank ℚ K : ℝ))
  have hB : 0 ≤ B := Real.rpow_nonneg_of_nonneg toReal_nonneg _
  have h_le : (minkowskiBound K) ≤ volume (convexBodySum K B) := by
    refine le_of_eq ?_
    rw [convexBodySum_volume, ← ENNReal.ofReal_pow hB, ← Real.rpow_nat_cast, ← Real.rpow_mul
      toReal_nonneg, div_mul_cancel _ (Nat.cast_ne_zero.mpr (ne_of_gt finrank_pos)), Real.rpow_one,
      ofReal_toReal, mul_comm, mul_assoc, ENNReal.inv_mul_cancel (convexBodySumFactor_ne_zero K)
      (convexBodySumFactor_ne_top K), mul_one]
    exact mul_ne_top (ne_of_lt (minkowskiBound_lt_top K))
      (ENNReal.inv_ne_top.mpr (convexBodySumFactor_ne_zero K))
  obtain ⟨x, h_nz, h_bd⟩ := exists_ne_zero_mem_ringOfIntegers_of_norm_le K h_le
  refine ⟨x, h_nz, ?_⟩
  convert h_bd
  rw [div_pow B, ← Real.rpow_nat_cast B, ← Real.rpow_mul (by positivity), div_mul_cancel _
    (Nat.cast_ne_zero.mpr <| ne_of_gt finrank_pos), Real.rpow_one, mul_comm_div, mul_div_assoc']
  congr 1
  rw [eq_comm]
  calc
    _ = (2:ℝ)⁻¹ ^ NrComplexPlaces K * sqrt ‖discr K‖₊ * (2:ℝ) ^ finrank ℚ K *
          ((2:ℝ) ^ NrRealPlaces K * (π / 2) ^ NrComplexPlaces K /
          (Nat.factorial (finrank ℚ K)))⁻¹ := by
      simp_rw [minkowskiBound, convexBodySumFactor, volume_fundamentalDomain_latticeBasis,
        toReal_mul, toReal_inv, toReal_div, toReal_mul, coe_toReal, toReal_pow, toReal_inv,
        toReal_ofNat, mixedEmbedding.finrank, toReal_div, toReal_ofNat, coe_toReal, coe_real_pi,
        toReal_nat]
    _ = (2:ℝ) ^ (finrank ℚ K - NrComplexPlaces K - NrRealPlaces K + NrComplexPlaces K : ℤ) *
          Real.sqrt ‖discr K‖ * Nat.factorial (finrank ℚ K) * π⁻¹ ^ (NrComplexPlaces K) := by
      simp_rw [inv_div, div_eq_mul_inv, mul_inv, ← zpow_neg_one, ← zpow_coe_nat, mul_zpow,
        ← zpow_mul, neg_one_mul, mul_neg_one, neg_neg, Real.coe_sqrt, coe_nnnorm, sub_eq_add_neg,
        zpow_add₀ (two_ne_zero : (2:ℝ) ≠ 0)]
      ring
    _ = (2:ℝ) ^ (2 * NrComplexPlaces K : ℤ) * Real.sqrt ‖discr K‖ * Nat.factorial (finrank ℚ K) *
          π⁻¹ ^ (NrComplexPlaces K) := by
      congr
      rw [← card_add_two_mul_card_eq_rank, Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat]
      ring
    _ = (4 / π) ^ NrComplexPlaces K * (finrank ℚ K).factorial * Real.sqrt |discr K| := by
      rw [show ‖discr K‖ = |(discr K : ℝ)| by rfl, zpow_mul, show (2:ℝ) ^ (2:ℤ) = 4 by norm_cast,
        div_pow, inv_eq_one_div, div_pow, one_pow, zpow_coe_nat]
      

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ := ENNReal.toReal (minkowskiBound K * (convexBodySumFactor K)⁻¹) ^ (1 / ↑(finrank ℚ K))
hB : 0 ≤ B
h_le : minkowskiBound K ≤ ↑↑volume (convexBodySum K B)
x : ↥(𝓞 K)
h_nz : x ≠ 0
h_bd : ↑|(Algebra.norm ℚ) ↑x| ≤ (B / ↑(finrank ℚ K)) ^ finrank ℚ K
⊢ 4 ^ NrComplexPlaces K * Real.sqrt |↑(discr K)| * ↑(Nat.factorial (finrank ℚ K)) * (1 / π ^ NrComplexPlaces K) =
    4 ^ NrComplexPlaces K / π ^ NrComplexPlaces K * ↑(Nat.factorial (finrank ℚ K)) * Real.sqrt |↑(discr K)|

Next tactic:
ring