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Proof step 10 from proof bertrand-estimate (previous step)

4 ^ (i / 3) ≤ 4 ^ (i - (2 * i) / 3), i / 3 ≤ i - (2 * i) / 3, (2 * i) / 3 ≤ i, 0 ≤ 4 ^ i, 4 ^ (i - (2 * i) / 3) ≤ (2 * i) ^ sqrt(2 * i), 0 ≤ i, 0 < i, 1024 ≤ i, 4 ^ i ≤ 4 ^ ((2 * i) / 3) * (2 * i) ^ sqrt(2 * i), 0 < 4 ^ ((2 * i) / 3), 0 ≤ (2 * i) ^ sqrt(2 * i), 0 ≤ (2 * i) / 3 ⊦

User applied rule cut formula

And obtained the following new goals

Proof step 11:
4 ^ (i / 3) ≤ (2 * i) ^ sqrt(2 * i), 4 ^ (i / 3) ≤ 4 ^ (i - (2 * i) / 3), i / 3 ≤ i - (2 * i) / 3, (2 * i) / 3 ≤ i, 0 ≤ 4 ^ i, 4 ^ (i - (2 * i) / 3) ≤ (2 * i) ^ sqrt(2 * i), 0 ≤ i, 0 < i, 1024 ≤ i, 4 ^ i ≤ 4 ^ ((2 * i) / 3) * (2 * i) ^ sqrt(2 * i), 0 < 4 ^ ((2 * i) / 3), 0 ≤ (2 * i) ^ sqrt(2 * i), 0 ≤ (2 * i) / 3 ⊦

Rule argument

fma: 4 ^ (i / 3) ≤ (2 * i) ^ sqrt(2 * i)

Additional information

Used simplifier rules: sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs Used formulas: 4 ^ (i / 3) ≤ (2 * i) ^ sqrt(2 * i)