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Proof step 13 from proof bertrand-estimate (previous step)

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 ⊦ 4 ^ (i / 3) ≤ 4 ^ (i - (2 * i) / 3)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-pot inst: name: le-pos seq: ⊦ 1 < i ∧ 0 ≤ j → (i ^ j ≤ i ^ j0 ↔ j ≤ j0) subst: [j0, j, i] → [i - (2 * i) / 3, i / 3, 4] flags: rewrite

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%pos: ⊦ 0 < j ∧ 0 ≤ i → 0 ≤ i / j; from specification int-div Used terms: i - (2 * i) / 3, i / 3, 4