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Proof step 4 from proof bertrand-estimate (previous step)

¬(0 < 4 ^ ((2 * i) / 3) ∧ 0 ≤ 4 ^ i ∧ ¬ 4 ^ i / 4 ^ ((2 * i) / 3) ≤ (2 * i) ^ sqrt(2 * i)), 4 ^ i ≤ 4 ^ ((2 * i) / 3) * (2 * i) ^ sqrt(2 * i), 1024 ≤ i, 0 < i, 0 ≤ i ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
0 ≤ 4 ^ i, 4 ^ i / 4 ^ ((2 * i) / 3) ≤ (2 * i) ^ sqrt(2 * i), 0 ≤ i, 0 < i, 1024 ≤ i, 4 ^ i ≤ 4 ^ ((2 * i) / 3) * (2 * i) ^ sqrt(2 * i), 0 < 4 ^ ((2 * i) / 3), 0 ≤ (2 * i) ^ sqrt(2 * i) ⊦

Rule argument

Additional information

Used simplifier rules: zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%notneg-02: ⊦ 0 < i → ¬ i ^ j < 0; from specification int-pot%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%div-mult-nls-zero: ⊦ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → ¬ (i * i0) / j < 0; from specification int-div%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%grz: ⊦ 0 < i → (0 < i ^ j ↔ i = 1 ∨ 0 ≤ j); from specification int-pot