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Proof step 16 from proof bertrand-estimate (previous step)

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

User applied rule insert elim lemma

And obtained the following new goals

Proof step 17:
i = i0 * 3 + j0 ∧ 0 ≤ j0 ∧ j0 < 3 ∧ 0 ≤ i0 ∧ 0 < 3, i < i0 + (2 * i) / 3, 0 ≤ (2 * i) / 3, 0 ≤ (2 * i) ^ sqrt(2 * i), 0 < 4 ^ ((2 * i) / 3), 4 ^ i ≤ 4 ^ ((2 * i) / 3) * (2 * i) ^ sqrt(2 * i), 1024 ≤ i, 0 < i, 0 ≤ i, 4 ^ (i - (2 * i) / 3) ≤ (2 * i) ^ sqrt(2 * i), 0 ≤ 4 ^ i, (2 * i) / 3 ≤ i, 0 < i0 + (2 * i) / 3, 1024 < i0 + (2 * i) / 3, 0 < i0 ⊦

Rule argument

spec: int-div inst: name: div-elim-pospos seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0) subst: [i, j, i0, j0] → [i, 3, i0, j] flags: []

Additional information

Used simplifier rules: zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0); (elim) Used terms: i, 3, i0, j