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Proof step 17 from proof bertrand-estimate (previous step)

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 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 18:
0 ≤ i0, 0 ≤ j0, 0 < i0, 0 < i0 + (2 * (i0 * 3 + j0)) / 3, 0 ≤ 4 ^ (i0 * 3 + j0), 0 < i0 * 3 + j0, 0 < 4 ^ ((2 * (i0 * 3 + j0)) / 3), 0 ≤ (2 * (i0 * 3 + j0)) ^ sqrt(2 * (i0 * 3 + j0)), 0 ≤ (2 * (i0 * 3 + j0)) / 3, 2 * i0 + j0 < (2 * (i0 * 3 + j0)) / 3, j0 < 3, 1024 < i0 + (2 * (i0 * 3 + j0)) / 3, (2 * (i0 * 3 + j0)) / 3 ≤ i0 * 3 + j0, 4 ^ (i0 * 3 + j0 - (2 * (i0 * 3 + j0)) / 3) ≤ (2 * (i0 * 3 + j0)) ^ sqrt(2 * (i0 * 3 + j0)), 1024 ≤ i0 * 3 + j0, 4 ^ (i0 * 3 + j0) ≤ 4 ^ ((2 * (i0 * 3 + j0)) / 3) * (2 * (i0 * 3 + j0)) ^ sqrt(2 * (i0 * 3 + j0)) ⊦

Rule argument

Additional information

Used simplifier rules: zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult%numlslsdel-01: i0 ≠ 0, i0 < i ⊦ (* i) < j → (* i0) < j; from specification int-basic1%ls-pos: ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); from specification int-mult%ls-01: ⊦ i + j < i + i0 ↔ j < i0; from specification int-basic2%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%zero-ge-add: ⊦ 0 ≤ j1 ∧ j1 < j → (0 ≤ i0 * j + j1 ↔ 0 ≤ i0); from specification int-mult%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%numls-03: ⊦ (* i2) * i + j < i + j0 ↔ (i2 - 1) * i + j < j0; from specification int-mult