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Proof step 19 from proof pot-root-est (previous step)

2 * i = j * j + j0, j = sqrt(2 * i), 4 ^ (j / 6 * j) < ((j + 1) ^ 2) ^ j, 2 * i < (j + 1) ^ 2, 0 < i, (2 * i) ^ j < ((j + 1) ^ 2) ^ j, j / 6 * j ≤ (j ^ 2 / 2) / 3, j ^ 2 ≤ 2 * i, 0 ≤ i, 4 ^ (i / 3) ≤ (2 * i) ^ sqrt(2 * i), 1024 ≤ i, j0 ≤ 2 * j, 0 ≤ j, 0 ≤ j0, j ^ 2 / 2 ≤ i, (j ^ 2 / 2) / 3 ≤ i / 3, 4 ^ (j / 6 * j) ≤ 4 ^ (i / 3) ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 20:
⊦ 0 ≤ j ∧ 0 ≤ j0 → i ^ (j * j0) = (i ^ j) ^ j0
Proof step 21:
2 * i = j * j + j0, j = sqrt(2 * i), (2 * i) ^ j < ((j + 1) ^ 2) ^ j, 0 < i, 2 * i < (j + 1) ^ 2, (4 ^ (j / 6)) ^ j < ((j + 1) ^ 2) ^ j, (4 ^ (j / 6)) ^ j ≤ 4 ^ (i / 3), (j ^ 2 / 2) / 3 ≤ i / 3, j ^ 2 / 2 ≤ i, 0 ≤ j0, j0 ≤ 2 * j, 1024 ≤ i, 4 ^ (i / 3) ≤ (2 * i) ^ sqrt(2 * i), 0 ≤ i, j ^ 2 ≤ 2 * i, j / 6 * j ≤ (j ^ 2 / 2) / 3, 0 ≤ j / 6, 0 ≤ j ⊦

Rule argument

spec: <toplevel> name: pot-pot-02 seq: ⊦ 0 ≤ j ∧ 0 ≤ j0 → i ^ (j * j0) = (i ^ j) ^ j0 subst: [j0, j, i] → [j, j / 6, 4] flags: <all paths>

Additional information

Used simplifier rules: j ≠ 0, j0 ≠ 0 ⊦ (i / (* j)) / (* j0) = i / (j * j0); (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ 0 < j ∧ 0 ≤ i → 0 ≤ i / j; (s)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s) Used terms: j, j / 6, 4