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Proof step 23 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, 0 ≤ j, 0 ≤ j / 6, 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 ≤ j0, j ^ 2 / 2 ≤ i, (j ^ 2 / 2) / 3 ≤ i / 3, (4 ^ (j / 6)) ^ j ≤ 4 ^ (i / 3) ⊦ 0 ≤ 4 ^ (j / 6) ∧ 0 ≤ (j + 1) ^ 2 ∧ 0 < j

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 24:
2 * i = sqrt(2 * i) * sqrt(2 * i) + j0, (sqrt(2 * i) + 1) ^ 2 < 0, 0 < sqrt(2 * i), (2 * i) ^ sqrt(2 * i) < ((sqrt(2 * i) + 1) ^ 2) ^ sqrt(2 * i), 0 < i, 2 * i < (sqrt(2 * i) + 1) ^ 2, (4 ^ (sqrt(2 * i) / 6)) ^ sqrt(2 * i) < ((sqrt(2 * i) + 1) ^ 2) ^ sqrt(2 * i), 4 ^ (i / 3) < ((sqrt(2 * i) + 1) ^ 2) ^ sqrt(2 * i), sqrt(2 * i) ^ 2 < (sqrt(2 * i) + 1) ^ 2, 0 ≤ 4 ^ (sqrt(2 * i) / 6), 0 ≤ sqrt(2 * i), (4 ^ (sqrt(2 * i) / 6)) ^ sqrt(2 * i) ≤ 4 ^ (i / 3), sqrt(2 * i) ^ 2 / 6 ≤ i / 3, sqrt(2 * i) ^ 2 / 2 ≤ i, 0 ≤ j0, j0 ≤ 2 * sqrt(2 * i), 1024 ≤ i, 4 ^ (i / 3) ≤ (2 * i) ^ sqrt(2 * i), 0 ≤ i, sqrt(2 * i) ^ 2 ≤ 2 * i, sqrt(2 * i) / 6 * sqrt(2 * i) ≤ sqrt(2 * i) ^ 2 / 6, (4 ^ (sqrt(2 * i) / 6)) ^ sqrt(2 * i) ≤ (2 * i) ^ sqrt(2 * i), sqrt(2 * i) / 6 * sqrt(2 * i) ≤ i / 3 ⊦

Rule argument

Additional information

Used simplifier rules: notneg-02: ⊦ 0 < i → ¬ i ^ j < 0;% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs% ⊦ 0 < i → ¬ sqrt(2 * i) ≤ 0; (s)% ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); (s)%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs% ⊦ 0 < j ∧ 0 ≤ i → 0 ≤ i / j; (s)%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%j ≠ 0, j0 ≠ 0 ⊦ (i / (* j)) / (* j0) = i / (j * j0); (s)