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Proof step 20 from proof bertrand-estimate (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: div-add1-pospos seq: ⊦ i % j = 0 ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j subst: [j, i0, i] → [3, j0 * 2, (i0 * 3) * 2] flags: rewrite

Additional information

Used simplifier rules: associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%mod-zero-num: i0 % j = 0, j ≠ 0 ⊦ (i * (* i0)) % (* j) = 0; from specification int-div%zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%div-mult-num: j ≠ 0, j0 % j = 0 ⊦ (i * (* j0)) / (* j) = i * j0 / j; from specification int-div%numsucc: ⊦ (* i0) * i + i = (i0 + 1) * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%ls-01: ⊦ i + j < i + i0 ↔ j < i0; from specification int-basic2%nls-mult-div: ⊦ 0 ≤ i ∧ 0 < j ∧ i1 < j → ¬ i < (i * i1) / j; from specification int-div Used terms: 3, j0 * 2, (i0 * 3) * 2