/-- A reified version of the `Bertrand.main_inequality` below.
This is not best possible: it actually holds for 464 ≤ x.
-/
theorem real_main_inequality {x : ℝ} (n_large : (512 : ℝ) ≤ x) :
    x * (2 * x) ^ sqrt (2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x := by
  let f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
  have hf' : ∀ x, 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3) := fun x h =>
    div_pos (mul_pos h (rpow_pos_of_pos (mul_pos two_pos h) _)) (rpow_pos_of_pos four_pos _)
  have hf : ∀ x, 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)) := by
    intro x h5
    have h6 := mul_pos (zero_lt_two' ℝ) h5
    have h7 := rpow_pos_of_pos h6 (sqrt (2 * x))
    rw [log_div (mul_pos h5 h7).ne' (rpow_pos_of_pos four_pos _).ne', log_mul h5.ne' h7.ne',
      log_rpow h6, log_rpow zero_lt_four, ← mul_div_right_comm, ← mul_div, mul_comm x]
  have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) n_large
  rw [← div_le_one (rpow_pos_of_pos four_pos x), ← div_div_eq_mul_div, ← rpow_sub four_pos, ←
    mul_div 2 x, mul_div_left_comm, ← mul_one_sub, (by norm_num1 : (1 : ℝ) - 2 / 3 = 1 / 3),
    mul_one_div, ← log_nonpos_iff (hf' x h5), ← hf x h5]
  -- porting note: the proof was rewritten, because it was too slow
  have h : ConcaveOn ℝ (Set.Ioi 0.5) f := by
    apply ConcaveOn.sub
    apply ConcaveOn.add
    exact strictConcaveOn_log_Ioi.concaveOn.subset
      (Set.Ioi_subset_Ioi (by norm_num)) (convex_Ioi 0.5)
    convert ((strictConcaveOn_sqrt_mul_log_Ioi.concaveOn.comp_linearMap
      ((2 : ℝ) • LinearMap.id))) using 1
    · ext x
      simp only [Set.mem_Ioi, Set.mem_preimage, LinearMap.smul_apply,
        LinearMap.id_coe, id_eq, smul_eq_mul]
      rw [← mul_lt_mul_left (two_pos)]
      norm_num1
      rfl
    apply ConvexOn.smul
    refine div_nonneg (log_nonneg (by norm_num1)) (by norm_num1)
    exact convexOn_id (convex_Ioi (0.5 : ℝ))
  suffices ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0 by
    obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this
    exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4
  refine' ⟨18, 512, by norm_num1, by norm_num1, n_large, _, _⟩
  · have : sqrt (2 * 18) = 6 := (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
    rw [hf, log_nonneg_iff, this]
    rw [one_le_div] <;> norm_num1
    apply le_trans _ (le_mul_of_one_le_left _ _) <;> norm_num1
    apply Real.rpow_le_rpow <;> norm_num1
    apply rpow_nonneg_of_nonneg; norm_num1
    apply rpow_pos_of_pos; norm_num1
    apply hf' 18; 

Proof state:
case refine'_1
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f
this : sqrt (2 * 18) = 6
⊢ 0 < 18
case refine'_1.a
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + sqrt (2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ sqrt (2 * x) / 4 ^ (x / 3))
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f
this : sqrt (2 * 18) = 6
⊢ 0 < 18

Next tactic:
norm_num1