-- Mathlib/NumberTheory/Bertrand.lean
theorem real_main_inequality {x : ℝ} (x_large : (512 : ℝ) ≤ x) :
    x * (2 * x) ^ √(2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Proof state:
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f
this : √(2 * 512) = 32
⊢ 0 < 512

Tactic:
norm_num1

Result:
no goals

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

Tactic:
norm_num1

Result:
no goals

Proof state:
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f
⊢ f x ≤ 0

Tactic:
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

Result:
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f
⊢ ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0

Proof state:
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
⊢ x * (2 * x) ^ √(2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Tactic:
have h5 : 0 < x := lt_of_lt_of_le (by norm_num1) x_large

Result:
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ x * (2 * x) ^ √(2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x

Proof state:
case refine_2
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f
this : √(2 * 512) = 32
⊢ 2 ^ (9 + 10 * 32) ≤ (2 ^ 2) ^ (512 / 3)

Tactic:
rw [← rpow_mul, ← rpow_natCast]

Result:
case refine_2
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f
this : √(2 * 512) = 32
⊢ 2 ^ ↑(9 + 10 * 32) ≤ 2 ^ (2 * (512 / 3))

case refine_2.hx
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f
this : √(2 * 512) = 32
⊢ 0 ≤ 2

Proof state:
case refine_2
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f
this : √(2 * 512) = 32
⊢ 2 ^ 9 * (2 ^ 10) ^ ↑32 ≤ 4 ^ (512 / 3)

Tactic:
rw [rpow_natCast, ← pow_mul, ← pow_add]

Result:
case refine_2
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f
this : √(2 * 512) = 32
⊢ 2 ^ (9 + 10 * 32) ≤ 4 ^ (512 / 3)

Proof state:
case refine_1
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f
this : √(2 * 18) = 6
⊢ 0 ≤ f 18

Tactic:
rw [hf _ (by norm_num1), log_nonneg_iff (by positivity), this, one_le_div (by norm_num1)]

Result:
case refine_1
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f
this : √(2 * 18) = 6
⊢ 4 ^ (18 / 3) ≤ 18 * (2 * 18) ^ 6

Proof state:
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f
this : √(2 * 512) = 32
⊢ 512 = 2 ^ 9

Tactic:
norm_num1

Result:
no goals

Proof state:
case refine_2
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f
⊢ f 512 ≤ 0

Tactic:
have : √(2 * 512) = 32 :=
  (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)

Result:
case refine_2
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f
this : √(2 * 512) = 32
⊢ f 512 ≤ 0

Proof state:
case hf
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ ConcaveOn ℝ (Set.Ioi 0.5) fun x => log x + √(2 * x) * log (2 * x)

Tactic:
apply ConcaveOn.add

Result:
case hf.hf
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ ConcaveOn ℝ (Set.Ioi 0.5) log

case hf.hg
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ ConcaveOn ℝ (Set.Ioi 0.5) fun x => √(2 * x) * log (2 * x)

Proof state:
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
⊢ 1 ≤ 4

Tactic:
norm_num1

Result:
no goals