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Proof step 21 from proof pot-root-est (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: inst: name: ls-pos-03 seq: ⊦ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i ^ j < i0 ^ j ↔ i < i0) subst: [i0, j, i] → [(j + 1) ^ 2, j, 4 ^ (j / 6)] flags: rewrite, current

Additional information

Used simplifier rules: notneg-02: ⊦ 0 < i → ¬ i ^ j < 0;%notneg: ⊦ 0 ≤ i ∧ 0 ≤ j → ¬ i ^ j < 0;%pos: ⊦ 0 < j ∧ 0 ≤ i → 0 ≤ i / j; from specification int-div%div-div-num: j ≠ 0, j0 ≠ 0 ⊦ (i / (* j)) / (* j0) = i / (j * j0); from specification int-div%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%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%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1%trans: ⊦ i < j ∧ j < j0 → i < j0; (forward) from specification int-basic1%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%succ-01: ⊦ i +1 = i + 1; from specification int-basic2%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%numls-03: ⊦ i + (* j) < (* j0) ↔ i < j0 - j; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%square-not-neg-03: ⊦ 0 < i → ¬ sqrt(2 * i) ≤ 0; from specification int-square Used terms: (j + 1) ^ 2, j, 4 ^ (j / 6)