Theorem: centralBinomialUpper_monotone (ExponentialRamsey/Prereq/Mathlib/Analysis/SpecialFunctions/ExplicitStirling.lean)

State before:
⊢ Antitone centralBinomialUpper
Tactic:
refine' antitone_nat_of_succ_le _
State after:
⊢ ∀ (n : ℕ), centralBinomialUpper (n + 1) ≤ centralBinomialUpper n

State before:
⊢ ∀ (n : ℕ), centralBinomialUpper (n + 1) ≤ centralBinomialUpper n
Tactic:
intro n
State after:
n : ℕ
⊢ centralBinomialUpper (n + 1) ≤ centralBinomialUpper n

State before:
n : ℕ
⊢ centralBinomialUpper (n + 1) ≤ centralBinomialUpper n
Tactic:
rw [centralBinomialUpper, centralBinomialUpper, _root_.pow_succ, ← div_div]
State after:
n : ℕ
⊢ ↑(centralBinom (n + 1)) * Real.sqrt (↑(n + 1) + 1 / 3) / 4 / 4 ^ n ≤
    ↑(centralBinom n) * Real.sqrt (↑n + 1 / 3) / 4 ^ n

State before:
n : ℕ
⊢ ↑(centralBinom (n + 1)) * Real.sqrt (↑(n + 1) + 1 / 3) / 4 / 4 ^ n ≤
    ↑(centralBinom n) * Real.sqrt (↑n + 1 / 3) / 4 ^ n
Tactic:
refine' div_le_div_of_le (by positivity) _
State after:
n : ℕ
⊢ ↑(centralBinom (n + 1)) * Real.sqrt (↑(n + 1) + 1 / 3) / 4 ≤ ↑(centralBinom n) * Real.sqrt (↑n + 1 / 3)

State before:
n : ℕ
⊢ ↑(centralBinom (n + 1)) * Real.sqrt (↑(n + 1) + 1 / 3) / 4 ≤ ↑(centralBinom n) * Real.sqrt (↑n + 1 / 3)
Tactic:
rw [div_le_iff, mul_assoc, mul_comm _ (_ * _), ← div_le_div_iff, mul_comm, mul_div_assoc,
  centralBinom_ratio, Nat.cast_add_one]
State after:
n : ℕ
⊢ 4 * ((↑n + 1 / 2) / (↑n + 1)) ≤ 4 * (Real.sqrt (↑n + 1 / 3) / Real.sqrt (↑n + 1 + 1 / 3))

case b0
n : ℕ
⊢ 0 < ↑(centralBinom n)

case d0
n : ℕ
⊢ 0 < Real.sqrt (↑(n + 1) + 1 / 3)

n : ℕ
⊢ 0 < 4

State before:
n : ℕ
⊢ 4 * ((↑n + 1 / 2) / (↑n + 1)) ≤ 4 * (Real.sqrt (↑n + 1 / 3) / Real.sqrt (↑n + 1 + 1 / 3))

case b0
n : ℕ
⊢ 0 < ↑(centralBinom n)

case d0
n : ℕ
⊢ 0 < Real.sqrt (↑(n + 1) + 1 / 3)

n : ℕ
⊢ 0 < 4
Tactic:
refine' mul_le_mul_of_nonneg_left (upper_monotone_aux _) (by positivity)
State after:
case b0
n : ℕ
⊢ 0 < ↑(centralBinom n)

case d0
n : ℕ
⊢ 0 < Real.sqrt (↑(n + 1) + 1 / 3)

n : ℕ
⊢ 0 < 4

State before:
n : ℕ
⊢ 0 ≤ 4 ^ n
Tactic:
positivity
State after:
no goals

State before:
n : ℕ
⊢ 0 ≤ 4
Tactic:
positivity
State after:
no goals

State before:
case b0
n : ℕ
⊢ 0 < ↑(centralBinom n)
Tactic:
rw [Nat.cast_pos]
State after:
case b0
n : ℕ
⊢ 0 < centralBinom n

State before:
case b0
n : ℕ
⊢ 0 < centralBinom n
Tactic:
exact centralBinom_pos _
State after:
no goals

State before:
case d0
n : ℕ
⊢ 0 < Real.sqrt (↑(n + 1) + 1 / 3)
Tactic:
positivity
State after:
no goals

State before:
n : ℕ
⊢ 0 < 4
Tactic:
positivity
State after:
no goals