open MeasureTheory MeasureTheory.Measure Real in
theorem convexBodySum_volume :
    volume (convexBodySum K B) = (convexBodySumFactor K) * (.ofReal B) ^ (finrank ℚ K) := by
  obtain hB | hB := le_or_lt B 0
  · rw [convexBodySum_volume_eq_zero_of_le_zero K hB, ofReal_eq_zero.mpr hB, zero_pow, mul_zero]
    exact finrank_pos
  · suffices volume (convexBodySum K 1) = (convexBodySumFactor K) by
      rw [mul_comm]
      convert addHaar_smul volume B (convexBodySum K 1)
      · simp_rw [← Set.preimage_smul_inv₀ (ne_of_gt hB), Set.preimage_setOf_eq, convexBodySumFun,
          Prod.smul_fst, Prod.smul_snd, Pi.smul_apply, Complex.real_smul, smul_eq_mul, norm_mul,
          Complex.ofReal_inv, norm_inv, norm_eq_abs B, Complex.norm_eq_abs B, Complex.abs_ofReal,
          abs_eq_self.mpr (le_of_lt hB), ← Finset.mul_sum, ← mul_assoc, mul_comm (2:ℝ), mul_assoc,
          ← mul_add, inv_mul_le_iff hB, mul_one]
      · rw [abs_pow, ofReal_pow (abs_nonneg _), abs_eq_self.mpr (le_of_lt hB),
          mixedEmbedding.finrank]
      · exact this.symm
    rw [MeasureTheory.measure_le_eq_lt _ ((convexBodySumFun_eq_zero_iff 0).mpr rfl)
      convexBodySumFun_neg convexBodySumFun_add_le
      (fun hx => (convexBodySumFun_eq_zero_iff _).mp hx)
      (fun r x => le_of_eq (convexBodySumFun_smul r x))]
    rw [measure_lt_one_eq_integral_div_gamma (g := fun x : (E K) => convexBodySumFun x)
      volume ((convexBodySumFun_eq_zero_iff 0).mpr rfl) convexBodySumFun_neg convexBodySumFun_add_le
      (fun hx => (convexBodySumFun_eq_zero_iff _).mp hx)
      (fun r x => le_of_eq (convexBodySumFun_smul r x)) zero_lt_one]
    simp_rw [mixedEmbedding.finrank, div_one, Gamma_nat_eq_factorial, ofReal_div_of_pos
      (Nat.cast_pos.mpr (Nat.factorial_pos _)), Real.rpow_one, ofReal_coe_nat]
    suffices ∫ x : E K, exp (-convexBodySumFun x) =
        (2:ℝ) ^ NrRealPlaces K * (π / 2) ^ NrComplexPlaces K by
      rw [this, convexBodySumFactor, ofReal_mul (by positivity), ofReal_pow zero_le_two,
        ofReal_pow (by positivity), ofReal_div_of_pos zero_lt_two, ofReal_ofNat,
        ← NNReal.coe_real_pi, ofReal_coe_nnreal]
    

Proof state:
case inr
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
hB : 0 < B
⊢ ∫ (x : ({ w // IsReal w } → ℝ) × ({ w // IsComplex w } → ℂ)), rexp (-convexBodySumFun x) =
    2 ^ NrRealPlaces K * (π / 2) ^ NrComplexPlaces K

Next tactic:
calc
      _ = (∫ x : {w : InfinitePlace K // IsReal w} → ℝ, ∏ w, exp (- ‖x w‖)) *
              (∫ x : {w : InfinitePlace K // IsComplex w} → ℂ, ∏ w, exp (- 2 * ‖x w‖)) := by
        simp_rw [convexBodySumFun, neg_add, ← neg_mul, Finset.mul_sum, ← Finset.sum_neg_distrib,
          exp_add, exp_sum, ← integral_prod_mul, volume_eq_prod]
      _ = (∫ x : ℝ, exp (-|x|)) ^ NrRealPlaces K *
              (∫ x : ℂ, Real.exp (-2 * ‖x‖)) ^ NrComplexPlaces K := by
        rw [integral_fintype_prod_eq_pow _ (fun x => exp (- ‖x‖)), integral_fintype_prod_eq_pow _
          (fun x => exp (- 2 * ‖x‖))]
        simp_rw [norm_eq_abs]
      _ =  (2 * Gamma (1 / 1 + 1)) ^ NrRealPlaces K *
              (π * (2:ℝ) ^ (-(2:ℝ) / 1) * Gamma (2 / 1 + 1)) ^ NrComplexPlaces K := by
        rw [integral_comp_abs (f := fun x => exp (- x)), ← integral_exp_neg_rpow zero_lt_one,
          ← Complex.integral_exp_neg_mul_rpow le_rfl zero_lt_two]
        simp_rw [Real.rpow_one]
      _ = (2:ℝ) ^ NrRealPlaces K * (π / 2) ^ NrComplexPlaces K := by
        simp_rw [div_one, one_add_one_eq_two, Gamma_add_one two_ne_zero, Gamma_two, mul_one,
          mul_assoc, ← Real.rpow_add_one two_ne_zero, show (-2:ℝ) + 1 = -1 by norm_num,
          Real.rpow_neg_one]
        rfl