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Proof step 41 from proof pot-root-est (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 42:
⊦ 1 < i ∧ 0 ≤ j → (i ^ j ≤ i ^ j0 ↔ j ≤ j0)
Proof step 43:
2 * i = j * j + j0, j = sqrt(2 * i), (2 * i) ^ j < ((j + 1) ^ 2) ^ j, 0 < i, 2 * i < (j + 1) ^ 2, 1 < 4, (j ^ 2 / 2) / 3 ≤ i / 3, j ^ 2 / 2 ≤ i, 0 ≤ j0, 0 ≤ j, j0 ≤ 2 * j, 1024 ≤ i, 4 ^ (i / 3) ≤ (2 * i) ^ sqrt(2 * i), 0 ≤ i, j ^ 2 ≤ 2 * i, j * j / 6 ≤ (j ^ 2 / 2) / 3, 0 ≤ j * j / 6 ⊦ j * j / 6 ≤ i / 3

Rule argument

spec: <toplevel> name: le-pos seq: ⊦ 1 < i ∧ 0 ≤ j → (i ^ j ≤ i ^ j0 ↔ j ≤ j0) subst: [j0, j, i] → [i / 3, 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% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ 0 ≤ i → (0 ≤ i * j ↔ i = 0 ∨ 0 ≤ j); (s)% ⊦ 0 < i → sqrt(2 * i) ≠ 0; (s)% ⊦ 0 < j ∧ 0 ≤ i → 0 ≤ i / j; (s) Used terms: i / 3, j * j / 6, 4