Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: case inr
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
hB : 0 < B
⊢ volume (convexBodySum K B) = ↑(convexBodySumFactor K) * ENNReal.ofReal B ^ finrank ℚ K

Tactic:
· 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_ofPred_eq, convexBodySumFun,
        normAtPlace_smul, abs_inv, abs_eq_self.mpr (le_of_lt hB), ← mul_assoc, mul_comm, mul_assoc,
        ← Finset.mul_sum, inv_mul_le_iff₀ 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 : (mixedSpace 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_natCast]
    suffices ∫ x : mixedSpace 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, coe_div (Nat.cast_ne_zero.mpr
        (Nat.factorial_ne_zero _)), coe_mul, coe_pow, coe_pow, coe_ofNat, coe_div two_ne_zero,
        coe_ofNat, coe_natCast]
    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_apply', 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_volume_eq_pow (fun x => exp (-‖x‖)),
          integral_fintype_prod_volume_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, div_eq_mul_inv]

Goal state after: